m    -  - 


BCAL 


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THE 

SIX-CHORD  SPIRAL 


BY 


J.   R.   STEPHENS,  C.E. 


NEW    YORK 

THE  ENGINEERING  NEWS   PUBLISHING  CO. 
1907 


-< 


< 


COPYRIGHT,  1907 

BY 
THE  ENGINEERING  NEWS  PUBLISHING  COMPANY 


PREFACE 

THE  Six-Chord  Spiral  is  an  ordinary  multiform 
compound  curve  of  six  arcs  of  equal  length,  whose 
degrees  of  curvature  increase  in  the  order  of  the 
natural  numbers,  and  so  arranged  that  the  seventh 
arc  always  exactly  coincides  with  the  main  circu- 
lar curve. 

As  herein  outlined  it  has  several  valuable 
features. 

1st.   It  is  perfectly,  flexible  and  always  fits. 

2d.  No  special  tables  whatever  are  required  for 
general  use.  Hence  such  tables  cannot  be  lost 
or  mislaid.  If  desired,  special  tables  of  the  usual 
form  may  be  quickly  computed  from  Table  IV  and 
formulas  (1)  and  (8). 

3d.  The  spiral  is  adapted  to  the  curve,  and  not 
the  curve  to  the  spiral  of  fixed  offset  or  length,  as 
is  the  case  with  table  spirals. 

4th.  Odd  curves  are  as  readily  fitted  as  even 
ones,  which  saves  time  and  trouble  in  spiraling 
old  track. 

5th.  Intermediate  transit  points  may  be  set 
at  any  plus,  and  do  not  lead  to  complex  deflec- 
tion calculations. 

6th.  The  method  is  quickly  grasped,  memorized, 
iii 

382062 


iv  PREFACE 

and  applied  by  transitmen  with  no  previous  knowl- 
edge of  spirals,  being  based  on  what  they  already 
know;  and  the  mathematical  treatment  being 
elementary  throughout. 

On  location  it  is  not  even  necessary  to  run  in 
the  six-chord,  a  terminal  curve  of  half  the  degree 
of  the  main  curve  and  giving  the  same  length  as 
the  spiral  line  being  substituted. 

In  this  connection  note  that  curves  are  usually 
traced  a  number  of  times  and  by  different  men 
before  the  final  centering. 

7th.  It  is  perfectly  interchangeable  with  the 
cubic  parabola,  the  two  being,  within  the  common 
limits  of  spiraling,  practically  identical. 

It  should  be  noted  that  no  spiral  changes  its 
degree  of  curvature  directly  with  the  elevation 
of  the  outer  rail,  when  the  elevation  approach  has 
vertical  curves  at  the  beginning  and  end.  In  this 
respect  all  spirals  are  misfits. 

The  importance  of  a  proper  length  of  spiral  is 
dwelt  upon,  and  methods  are  given  to  insure  con- 
sistency in  this  respect  with  varying  conditions 
of  speed  and  curve. 

Comparisons  are  made  between  spirals  commonly 
used,  which,  with  the  same  conditions,  define  their 
relations,  not  only  in  length  and  total  angle,  but 
also  laterally. 

The  second  part  deals  with  methods  for  shifting 
old  tracks  to  make  room  for  spirals,  pointing  out 


PREFACE  v 

that  this  question  is  entirely  independent  of  the 
kind  of  spiral  used. 

Acknowledgment  is  due  to  Professor  Talbot  for 
the  method  of  swinging  tangents  to  make  room 
for  spirals,  and  also  the  method  of  formulas  (27) 
and  (28)  for  inserting  a  spiral  between  the  two 
arcs  of  a  compound  curve  (see  Talbot's  "Transi- 
tion Spiral"). 

J.  R.  STEPHENS. 

DENVER,  COLORADO, 

November  5,  1906. 

NOTE.  — 

Natural  versed  sines  are  much  used  in  this  book. 

When  not  given  in  the  ordinary  field  tables,  they  may  be 
found  by  mentally  subtracting  the  natural  cosine  of  the  given 
angle  from  .9999  (10),  working  from  left  to  right,  and  calling 
the  last  decimal  used  10. 

To  find  the  angle  corresponding  to  a  given  natural  versed 
sine,  subtract  the  latter  from  .9999  (10),  as  above.  The 
remainder  will  be  the  natural  cosine  of  the  required  angle. 


CONTENTS 

PART    I 

LOCATION  AND  CONSTRUCTION  OF  SPIRALS 

PAGE 

General  Forms  of  Spirals 1 

The  Six-Chord  Spiral 4 

The  Six-Chord  Spiral  and  Terminal  Curve  Having  a 

Radius  Twice  that  of  Main  Curve     ....  7 
Formula  for  Substituting  Spirals  between  Two  Curves 
by   Shifting   Original   Tangent  —  Main    Curves 

Undisturbed        12 

Compound  Curves 14 

To  Shift  the  Two  Members  of  a  Compound  Curve  so 

that  Suitable  Spirals  may  be  Inserted     .      .      .17 

The  Length  of  Spirals 22 

The  Length  of  Spirals  Joining  Compound  Curves     .  25 

To  Run  in  the  Six-Chord  Spiral  by  Deflections     .      .  26 

The  Track  Parabola 29 

Relative  Lengths  and  Total  Angles  of  Spirals       .      .  31 

Demonstration  of  the  Six-Chord  Spiral     .      .      .      .  32 
Comparative  Tabulations :  Relations  between  the  Six- 
Chord  Spiral  and  Terminal  Curve,  Each  Being 

Exactly  and  Independently  Calculated     ...  35 

Comparison  of  Spirals  and  Summary 40 

PART  II 

SPIRALING  OLD  TRACK 

Methods 46 

Compound  Curves 53 

Space-Shifts  Preserving  Original  Length  of  Line         ,     59 

vii 


THE  SIX-CHORD   SPIRAL. 

PART   I. 

LOCATION  AND  CONSTRUCTION  OF 
SPIRALS. 

There  are  two  general  forms  of  spirals  in  com- 
mon use. 

1st.  The  Track  Parabola,  in  which  the  deflec- 
tions from  the  point  of  spiral  vary  as  the  squares 
of  the  distances  measured  from  the  same  point 
along  the  curve. 

With  the  track  parabola,  any  given  values  of 
EM  and  p,  Fig.  1,  are  fitted  exactly. 

Further,  any  intermediate  point  can  be  set 
exactly,  and,  the  instrument  being  moved  up, 
work  continued  in  a  manner  similar  to  that  used 
in  laying  out  circular  curves. 

This,  however,  sometimes  results  in  trouble  for 
inexperienced  men. 

2d.  The  Polychord  Spiral,  in  which  the  degree 
of  curve  increases  with  each  chord,  in  arithmetical 
progression. 

The  polychord  spiral  with  an  infinite  number 
of  chords  is  the  track  parabola. 

Reduced  to  its  simplest  form,  the  polychord  be- 
comes what  might  be  called  a  One-Chord  Spiral. 

The  latter  is  a  terminal  circular  curve  having  a 
radius  2  RM  (see  dotted  curve,  Fig.  1). 

1 


2  THE  SIX-CHORD  SPIRAL 

The  values  of  p  and  RM  being  fixed,  all  poly- 
chord  spirals  will  fall  between  the  one-chord  spiral 


and  the  track  parabola,  and  the  greater  the  number 
of  chords,  the  nearer  the  approach  to  the  track 
parabola. 


THE  SIX-CHORD  SPIRAL  3 

For  fixed  values  of  p  and  RM,  each  form  of 
spiral  has  its  own  appropriate  length,  the  one- 
chord  being  the  shortest  and  the  track  parabola 
the  longest,  all  the  poly  chords  falling  in  between; 
the  greater  the  number  of  chords  the  longer  the 
spiral. 

In  practice,  the  maximum  lateral  variation  of  a 
six-chord  from  a  parabola  will  not  exceed  0.02 
feet.  The  usual  variation  is  negligible  in  this 
class  of  work.  Hence  the  principal  easement 
curves  in  use  yield  alinements  which  approach 
each  other  so  closely  that  their  riding  qualities 
are  the  same. 

The  total  length  of  track,  between  common 
points  on  the  main  tangent  and  main  curve,  is 
also  the  same,  no  matter  what  spiral  be  used,  so 
that,  after  track  is  laid  to  a  one-chord,  it  may  be 
thrown  into  a  track  parabola  without  altering  the 
expansion. 

The  three  principal  classes  of  polychords  are: 

1st.  With  deflections  constant,  while  chord 
length  and  number  of  chords  vary  (such  as  the 
Searles  form). 

2d.  With  chord  length  constant,  while  deflec- 
tions and  number  of  chords  vary. 

3d.  With  number  of  chords  constant,  while  de- 
flections and  chord  lengths  vary. 

Most  of  these  spirals  depend  for  their  usefulness 
on  specially  prepared  tables,  which  must  be  con- 


4  THE  SIX-CHORD  SPIRAL 

suited  in  the  field,  and  their  efficiency  for  varying 
values  of  p  and  RM  increases  with  the  number  of 
tables. 

Thus,  Searles  has  provided  500  tabulated  spirals 
from  which  to  "select  the  one  coming  nearest  to 
given  values  of  p  and  RM- 

The  spiral  used  in  the  following  discussion  is  of 
the  third  type  and  has  invariably  six  chords. 

The  Six-Chord  Spiral  is  chosen: 

1st.  On  account  of  its  extremely  simple  rela- 
tion to  the  one-chord  spiral  or  terminal  arc  of  half 
the  degree  of  the  main  curve  (see  Fig.  2). 

2d.  On  account  of  its  close  approximation  to 
the  track  parabola,  and  all  polychords  commonly 
used. 

It  will  first  be  considered  as  a  curve  to  be  offset 
from  the  one-chord  spiral. 

The  offsets  are  small,  and  may  usually  be  esti- 
mated in  a  manner  analogous  to  the  use  of  the 
self-reading  rod  in  leveling. 

The  instrument  is  to  be  kept  on  the  one-chord 
spiral,  and  all  calculations,  shifts,  etc.,  are  made 
by  the  ordinary  rules  and  tables  for  circular  curves. 

Notes  are  kept  and  plats  made  precisely  as  for 
compound  curves. 

The  one-chord  is  sufficiently  exact  for  right-of- 
way  descriptions. 

Since  the  one-chord  and  the  six-chord  have  the 
same  length  between  common  points,  no  equation 


THE  SIX-CHORD  SPIRAL  5 

of  distance  is  introduced  in  passing  from  one  to 
the  other. 

To  aid  the  eye  in  offsetting  in  the  field  of  view 
of  the  instrument,  a  2J-inch  wrought-iron  washer 


FIG.  2. 

may  be  put  on  the  transit  rod.  This  will  give  a 
0.1  it.  offset  on  each  side  of  the  center  of  rod, 
which  is  usually  a  sufficient  help  for  setting  stakes. 
A  more  exact  makeshift  may  be  obtained  as 
follows : 


6  THE  SIX-CHORD  SPIRAL 

Take  a  two-foot  rule,  cut  off  the  two  outside 
hinged  legs,  thus  leaving  the  pivot  joint  with  a 
six-inch  leg  on  each  side.  Screw  one  of  these  legs 
along  a  face  of  an  ordinary  wooden  octagon  rod. 

The  other  leg  will  make  a  folding  offset  sight. 
This  movable  leg  should  have  fastened  to  its  face 
a  strip  of  sheet-iron,  say  6  in.  long  and  1  in.  wide, 
in  which  F-shaped  notches  are  cut,  deep  ones  for 
the  full  tenths  from  rod  center,  and  shallow  for 
the  half  tenths. 

When  the  vertical  hair  cuts  the  scale  at  the 
proper  offset,  set  tack  at  point  of  rod. 

In  case  the  spiral  is  so  long  that  a  division  into 
six  parts  gives  too  great  a  distance  between  track 
centers,  it  may  be  divided  into  twelve  equal  parts 
by  taking  every  fifth  point  in  Table  I. 

This  will  not  constitute  the  regular  twelve-chord 
spiral,  which  would  be  longer  and  include  a  greater 
total  angle  than  the  six-chord. 

As  a  guide  to  section  foremen  in  determining 
track  elevation  it  is  preferable  to  divide  the  spiral 
into  some  fixed  number  of  equal  parts,  regardless 
of  the  full  stationing. 

FORMULAS  (see  Fig.  2). 

p  =  RM(l-  cos  7\)  (1) 


cos  Ti  -     u-  (3) 


THE  SIX-CHORD  SPIRAL 


(5) 
(6) 


The  inferiors  "M,"  "1,"  and  "6"  indicate  re- 
spectively "main  curve,"  "one-chord,"  and  "six- 
chord."  L  and  R  are  lengths  of  arc  and  radius 
in  feet,  and  Dx  =  degree  of  one-chord. 

THE   SIX-CHORD   SPIRAL  AND  TERMINAL  CURVE 
HAVING  A  RADIUS  TWICE  THAT  OF  MAIN  CURVE. 

This  spiral  (Fig.  2)  has  six  chords,  each  one- 
fourth  length  of  terminal  curve,  hence  spiral  is 
1J  times  length  of  terminal  curve,  and  the  quarter 
points  H^H2H^H.>  of  the  terminal  curve,  are 
abreast  the  one-sixth  points  St  S2  $4  S5  of  the  spiral. 
$3  and  H3  coincide.  S3A  =  SSB.  One-half  the 
terminal  curve  is  inside  the  spiral,  the  other  half 
outside;  and  the  offsets  between  them,  at  equal 
distances  from  H3  or  S3,  are  equal.  H1  St  =  H5  S5 
=  .036  p,  and  H2  S2  =  H4  S4  =  .054  p.  The  offset 
p  -  EM  (1  -  cos  7\)  =  RM  X  versed  sine  T19 
where  RM  =  radius  of  main  curve,  and  Tl  =  the 

terminal  angle.     (1)   RM  =  ^ 


8  THE  SIX-CHORD  SPIRAL 

To  locate  the  spiral,  take  the  distance  for  gain- 
ing the  required  elevation  =  L6  =  6  C  (at  the 
nearest  multiple  of  six  feet,  to  avoid  fractional 
chaining) . 

Here  C  =  chord,  and  L6  =  6  C  =  length  of  spiral. 

Then  2  C  X  DM 

' 


where 


100 

DM  =  degree  of  main  curve. 
7\    =  terminal  angle  in  degrees. 
C     =  length  of  chord  in  feet. 


Next  calculate  p  from  equation   (1)   above  - 
run  in  the  terminal  curve  and    offset  to  spiral. 
Locate  P.  S.  and  $6,  on  outer  tangent  and  main 
curve,  one  chord-length  from  H^  and  H&  respect- 
ively. 

NOTE.—  TV  the  total  angle  of  six-chord  =  1£  TV 

Note  particularly  that  the  length  of  six-chord 
=  L6  is  1.5  times  the  length  of  the  one-chord  =  Lx; 
also,  as  an  aid  to  the  memory,  that  the  offset  .054 
=  1.5  times  .036. 

In  practice,  taking  p  at  4  feet,  the  offsets  would 
be  4  times  .054  =  0.216  ft.,  and  4  times  .036  = 
0.144  ft. 

Example.  —  Take  a  14°  curve  having  a  spiral 
approach  of  six  chords,  each  25  ft.  long  or  150  ft. 
in  all,  to  connect  with  a  7°  approach,  and  calcu- 
late the  offsets  to  spiral. 

RM  =  5730  *  14  =  409.3.     L6  =  25  X  6  =  150. 


THE  SIX-CHORD  SPIRAL  9 

7\  (the  terminal  angle)  =  J  L6  X  Z>  =  150  X 
14  -r-  3  =  7°,  L6  being  expressed  in  one  hundred- 
foot  units. 

The  main  offset  p  =  RM  (1  -  cos  7\)  =  409.3  X 
.00745  =  3.05  ft. 

The  offsets 

HA  =  H5S5  =  3.05  X  .036  =  0.11  ft. 
H2S2  =  HjS4  =  3.05  X  .054  =  0.16  ft. 
H3S3  =  Zero.    . 

The  P.  S.  and  S6  are  set  as  shown  in  Fig.  2. 

The  7°  approach  from  Hl  to  H5,  or  the  one- 
chord  spiral,  will  be  four  25-ft.  chords. 

Whenever  intermediate  offsets  are  required,  as 
in  centering  trestle  bents,  etc.,  the  following  table 
is  used: 


10 


THE  SIX-CHORD  SPIRAL 


TABLE  L 

TABLE  FOR  INTERMEDIATE  OFFSETS  TO  SIX-CHORD  SPIRAL 

FROM  MAIN  TANGENT  AND  MAIN  CURVE  WITH  ONE-CHORD 

APPROACH. 
(To    BE    MEASURED    INWARD    FROM    THE   MAIN    TANGENT 

HALF  OF  SPIRAL  AND  OUTWARD  FROM  THE  MAIN  CURVE 

HALF). 


"So 

*• 

to 

1 

*• 

bO 

5 

*• 

H 

a 

r£ 

•sl 

a 

1 

1 

|l 

a 

o 

1 

•§1 

I 

1 

TJ 

2  a 

"S 

t-<*H 

"g 

^    fl 

? 

Mt« 

"S 

l3 

"S 

t,«4H 

o 

'^'« 

o 

o  o  .a 

o 

^"^ 

o 

o  o  ^a 

o 

p 

o 

C    ,  — 

t 

-£ 

5« 

ji 
o 

S^l 

J3 
o 

5  "8 

1 

«-5g 

"3 

Sic 

*o 

*s 

Ste 

0 

o"g_« 

O 

§to 

"o 

o'B  m 

02 

•S  o 
£  « 

J3 

43 

«C 

n 

•so 
5fi  « 

03 

III 

,£) 

•50 
5fi? 

03 

— 

||| 

^ 

S 

«>.£ 

0M 

a 
£ 

S^ 

J 

1 

1^1 

1 

J.S 
M 

1 

P.S. 

.000 

S6 

Si 

.036 

S5 

S2 

.054 

S4 

.0000 

.0006 

.0004 

1 

.000 

9 

1 

.042 

9 

1 

.050 

9 

.0001 

.0006 

.0004 

2 

.001 

8 

2 

.048 

8 

2 

.046 

8 

.0002 

.0004 

.0005 

3 

.003 

7 

3 

.052 

7 

3 

.041 

7 

.0003 

.0004 

.0005 

4 

.006 

6 

4 

.056 

6 

4 

.036 

6 

.0003 

.0002 

.0005 

5 

.009 

5 

5 

.058 

5 

5 

.031 

5 

.0004 

.0001 

.0005 

6 

.013 

4 

6 

.059 

4 

6 

.026 

4 

.0005 

.0000 

.0006 

7 

.018 

3 

7 

.059 

3 

7 

.020 

3 

.0005 

.0000 

.0006 

8 

.023 

2 

8 

.059 

2 

8 

.014 

2 

.0006 

.0002 

.0007 

9 

.029 

1 

9 

.057 

1 

9 

.007 

1 

.0007 

.0003 

.0007 

Si 

.036 

S5 

S2 

.054 

S4 

S3 

.000 

S3 

Example.  —  In  the  preceding  example  let  the 
P.  S.  be  at  station  7  +  07,  chords  25  feet;  required 
the  offset  at  the  even  station  8. 

The  curve  may  be  tabulated  thus: 


THE  SIX-CHORD  SPIRAL 


11 


p.  S.  =  7  +  07 

51  =  7  +  32 

52  =  7  +  57 

53  =  7  +  82 
£4  =  8  +  07  J 


Hence  8  =  S3  +  if  =  S3  +  0.72 
toward  S4,  which,  by  interpolation 
in  Table  I,  equals  .042;  and 
.042  X  p  or  3.05  =  .128  ft. 


If  the  numbering  ran  in  the  opposite  direction, 
the  offset  at  6  +  40  being  required,  then  : 

p    q     _   n  _(_  C\<-7 


S  =  6  +  82 
'  _  „      __ 


Here  6  +  40  =  S3+  A  =  S3  +  0.32 
towardS2,which,byTableI,  equals 
-021  X  3.05  =  .064  ft. 


In  case  a  simple  curve  has  been  run  in  connect- 
ing the  main  tangents,  as  in  Fig.  1,  no  provision 
being  made  for  spiraling,  the  circular  curve  is 
moved  inward,  without  altering  the  original 
radius,  along  the  line  BC,  for  the  distance  EF  = 
p  -=-  cos  J  7,  where  p  is  the  principal  offset  and  / 
the  total  angle  turned  between  tangents,  EF 
being  parallel  to  BC.  Also  EG  =  p  tan  \  I. 

The  distance  back  to  the  P.  C.  from  G  of  the 
one-chord  spiral  approach  at  H  is  (see  Fig.  1) 


and 


GH 
EH 
AH 


RM  sin  7\ 

RM  sin  T1  +  p  tan 

(RM  +  p)  tan  J  /  - 


(8) 

I  (9) 

flM  sin  7\    (10) 


In  order  to  avoid  small  equations  and  to  fit  the 
ground  from  the  start,  the  one-chord  spiral  should 
be  run  in  on  the  first  located  line  that  is  likely  to 
become  final. 


12  THE  SIX-CHORD  SPIRAL 

FORMULA  FOR  SUBSTITUTING  SPIRALS  BETWEEN 
Two  CURVES,  BY  SHIFTING  THE  POSITION  OF 
THE  ORIGINAL  TANGPJNT  TO  MAKE  ROOM  FOR 
THE  SPIRALS,  LEAVING  MAIN  CURVES  UNDIS- 
TURBED. 

Let  A  be  the  angle  between  the  old  and  new 
tangents ; 

L  =  length  of   original    tangent;   pl  and   p2  = 
values   of   principal  offsets   selected   for  the  two 
curves  respectively; 

Ri  and  R2  =  radii  of  the  two  curves  respectively. 
Then,  when  the  curves  are  in  opposite  directions, 

A  (in  minutes)  =  344°   (?'  +  ft)  + 

/3440  (p  +  P2)V  x   OOQ145  (B.  +  B.). 
\  L/  /  L 

and  when  the  same  curves  are  in  the  same  direction, 

3440    (p,  -  TO 


A  (in  minutes)  = 


L 
X  .000145 


L 
Example.  —  Given  alinement  as  follows : 

Zero  =  P.  C.     9°  R  for  36°. 
4  =  P.  T. 

7  =  P.  C.    6°  L  for  30°. 
12  =  P.  T. 

To  insert  spirals  between  the  curves: 


THE  SIX-CHORD  SPIRAL  13 

By  Rule  1,  page  26,  for  length  of  spiral,  with 
speed  at  33J  miles  per  hour,  the  six-chord  for 
9°  =  331  x  6  in.  elevation  =  200  ft.;  and  six- 
chord  for  6°  =  33£  X  4  in.  elevation  =  133.3  ft. 

The  lengths  of  terminal  curves  are: 

133.3  of  4°  30'  for  9°  =  6°,  total  angle. 

88.9  of  3°  for  6°  =  2°  40',  total  angle. 

R,  +  R2  =  955  +  637  =  1592. 

637  X  vers  6°  =  3.49  -  ft. 

955  X  vers  2°  40'  =  1.03  =  ft,  and  ft  +  ft  = 
4.52. 

Then,  by  above  formula: 


A  =  - ^^2  +  f344°*4'52Y  X  .000145  X  1592 


300          '  V       300       J  '  300 

(The  original  tangent  being  300  feet  long), 
A  =  51.83'  +  2.07'  =  53.9'  =  54',  approx. 

This  is  10  feet  on  9°  curve,  and  15  feet  on  6°, 
and  the  corrected  alinement  without  terminal 
curves  would  read: 

Zero  =  P.  C.    9°  R  =  36°  54'. 
4  +  10  =  P.  T. 

6  +  85  =  P.  C.    6°  L  =  30°  54'. 
12  +  00  =  P.  T. 

Then,  as  one-half  of  each  terminal  curve  lies 
either  way  from  the  P.  T.  of  9°  and  the  P.  C.  of  6°, 
the  new  alinement  (ignoring  the  small  equation 
which  should  be  made  to  fall  on  the  new  tangent 
between  the  spirals)  will  be : 


14  THE  SIX-CHORD  SPIRAL 

Zero  =  P.O.     9°  R  for  30°  54',  total  angle. 

3  +  43.3  =  P.C.C.    4°  30'  R  for  6°,  total  angle. 

4  +  76.7  =  P.T. 

6  +  40.6  =  P.C.    3°  L  for  2°  40',  total  angle. 

7  +  29.5  =  P.C.C.    6°  L  for  28°  14',  total  angle. 
12  +  00     =  P.T. 

COMPOUND  CURVES. 

Whenever  the  degrees  of  curvature  of  the  two 
members  of  a  compound  curve  differ  materially, 
they  should  be  connected  by  a  spiral. 

This  spiral  should  be  run  in  on  the  original  loca- 
tion, to  save  the  trouble  of  subsequent  shifts, 
equations,  etc. 

The  general  method  before  described,  of  offsets 
from  a  one-chord  to  a  six-chord  spiral,  may  be 
applied  equally  well  in  this  case. 

The  one-chord  connection  averages  the  degrees 
of  the  adjacent  main  curves. 

Thus,  a  4°  compounding  into  an  8°  will  have  a 
one-chord  connection  of  J  (8  +  4)  =  6°. 

To  make  room  for  this  intermediate  6°,  a  suffi- 
cient offset  between  the  two  main  curves  must  be 
allowed,  and  the  sharper  curve  must  lie  inside  the 
lighter  one. 

The  length  of  the  one-chord  spiral,  the  principal 
offset  or  gap  p,  and  the  intermediate  offsets,  are 
determined  as  follows: 

Take  the  4°,  6°,  and  8°  combination  and  assume 


THE  SIX-CHORD  SPIRAL 


15 


that  the  whole  curvature  is  uniformly  "bent" 
outward  until  the  4°  becomes  a  tangent,  the  6°  a 
2°,  and  the  8°  a  4°. 

We  then  have  the  conditions  of  a  4°  curve  from 
tangent,  and  the  necessary  calculations  are  made, 
as  before  shown,  to  fit  these  conditions. 


P.  S.  to  S,  =  S5  to  SG  =  i  SA  -  i  H,H5 
AS3  =  AH,  =  BS3  -  BH3.    H,H3  =  H3H5 

NOTE.  —  All  "H"  points  are  on  one-chord  spiral;  all  "S" 
points  are  on  six-chord  spiral. 

Example.  —  (See  Searles'  "  R.  R.  Spiral,"  page  63, 
Art.  55.) 

Given  a  compound  curve  in  which  df  =  6°  and 
d"  =  10°  40',  to  replace  the  P.  C.  C.  by  a  spiral 
having  six  chords  of  25  ft.  each  (P.  S.  to  £6,  Fig.  3). 


16  THE  SIX-CHORD  SPIRAL 

First  determine  the  data  for  the  one-chord, 
#A,  Fig.  3. 

Its  degree  dt  =  J  (10°  40'  +  6°)  -  8°  20'. 

Its  length  I,  =  4  X  25  =  100  ft. 

Its  total  angle  /t  =  8J  X  100  =  8°  20',  of  which 
d'  X  i  /,  =  6°  X  .50  =  3°  is  deducted  from  the  6°, 
and  d"  X  K  =  10°  40'  X  .50  =  5°  20'  is  deducted 
from  the  10°  40'. 

The  total  angle  of  the  six-chord  spiral  will  be 

8°  20'  X  1.5  =  12°  30';  of  this,  d'  X  j 1,  =  6°  X  .75 

-  4°  30'  is  deducted  from  the  6°,  and  d"  X  J  I, 

=  10°  40'  X  .75  =  8°  is  deducted  from  the  10° 

40'. 

Note  that  in  this  case  the  choice  of  a  six-chord 
spiral  in  Searles  is  accidental.  The  above  rea- 
soning would  not  obtain  had  any  other  chord 
number  been  chosen. 

Now,  assuming  as  before  that  the  6°  curve  (the 
lightest  of  the  three)  be  bent  straight,  the  8°  20' 
curve  becomes  a  2°  20',  and  the  10°  40'  becomes  a 
4°  40'. 

Hence  the  conditions  are  a  2°  20'  one-chord 
approach  from  tangent  to  a  4°  40'  main  curve. 

The  terminal  angle  for  100  feet  of  2°  20'  curve 

=  2°  20',    and    p,  =  .436  X  2.33  X  1  =  1.02   [see 

(4),  page  9]:  or  p1  =  1228  X  .00083  =  1.02  [see 

(1),  page  8],  which  is  the  value  given  by  Searles, 

page  65. 

Then  with  the  instrument  at  H^  or  H5  (each 


THE  SIX-CHORD  SPIRAL  17 

being  two  chord  lengths  or  50  feet  from  the  middle 
point  S3  or  H3),  run  in  the  8°  20'  one-chord  spiral 
and  offset. 

HA  =  H5S5  =  1.02  X  .036  =  .037  ft. 
HA  =  H,S,  =  1.02  X  .054  -  .055  ft. 

Intermediate  offsets  are  interpolated  from  Table 
I,  as  before  shown. 

Since  in  this  particular  case  the  maximum  differ- 
ence between  the  one-chord  and  six-chord  is  but 
f  in.,  the  six-chord  might  well  be  omitted  until 
it  comes  to  the  final  adjustment  of  the  track. 

Note  the  direction  of  the  offsets,  outward  from 
the  one-chord  line  on  sharper  curve  half,  and  in- 
ward on  lighter  curve  half. 

Similarly  to  the  above,  the  length  of  the  one- 
chord  when  P!  is  given  may  be  determined  from 
formulas  3,  5  and  6,  page  9,  taking  4°  40'  as  the 
main  curve. 

To  SHIFT  THE  Two  MEMBERS  OF  A  COMPOUND 
CURVE  so  THAT  SUITABLE  SPIRALS  MAY  BE  IN- 
SERTED. 

Let  LEF,  Fig.  4,  be  a  compound  curve,  with  B 
and  C  as  centers  (b  and  c  being  the  total  angles), 
which  has  been  run  in  without  provision  for  spirals. 

Required  to  insert  spirals  without  changing  the 
degree  of  either  branch  of  the  original  compound. 

The  required  offsets  p  and  P,  Fig.  4,  are  to  be 


18 


THE  SIX-CHORD  SPIRAL 


taken  for  spirals  having  a  length  suitable  for  the 
speed  and  elevation  proposed. 


Assume  that  the  curve  EF  is  slid  inward,  along 
the  radial  line  EB  common  to  both  curves,  until 
F  falls  on  G,  E  on  D,  and  C  on  C7 


THE  SIX-CHORD  SPIRAL  19 

p 

Then  FG,  parallel  and  equal  to  ED  =  -  ,  where 

cose 

P  =  GN  and  angle  FGN  =  c;  also  FN  =  P  tan  c. 
Next  determine  the  proper  offset  pl  for  a  one- 
chord  JK  uniting  the  two  members  of  the  com- 
pound (see  Fig.  3  and  following). 

Then  EH  =  ED  -  Pl  =  -        -  p,.  (11) 


Assume  that  the  curve  EL  is  moved  inward 
until  E  falls  on  H  and  L  on  M,  EH  being  equal 
and  parallel  to  ML. 

Since  angle  TML  =  6,  M77  =  AfL  cos  6;  hence 


cos  c 


-Pl}cosb.  (12) 

^  ) 


If  the  curve  had  been  thus  run  in,  the  P.  T.  at 
M  would  be  a  distance,  M  U,  too  far  out  to  fit  the 
spiral  selected,  whose  principal  offset  is  p. 

To  make  this  fit,  the  provisional  P.  C.  at  G  must 
be  pushed  ahead  along  YG  produced,  for  a  dis- 
tance 

(-        -  p,J  cos  b  —  p 

GV  =  VCQSC  .    ,/  -  -  -  -  WN.    (13) 
sin  (b  +  c) 

If  (b  +  c)  exceeds  90°,  its  sine  will  be  sin  [180°  - 

9  +  en 

FW  =  P  tan  c  -  WN.  (14) 

From  W  add  the  distance  back  to  S,  making 


20  THE  SIX-CHORD  SPIRAL 

W S  =  CF  X  sin  7\,  where  R^  =  2  CF  (see  also 
equation  (8)  page  13). 

The  whole  curve,  with  one-chord  spirals,  may 
now  be  run  in,  remembering  to  deduct  from  the 
total  angle  c  the  terminal  angle  of  its  spiral  'to 
tangent  plus  the  angle  KCfD  of  the  one-chord 
KJ. 

Similarly,  the  total  angle  b  is  reduced  by  its  ter- 
minal spiral  angle  plus  the  angle  JB'H. 

Angle  KC'D  =  i  JK  X  degree  of  curve  EF. 
Angle  JB'H  =  \  JK  X  degree  of  curve  EL. 

In  the  case  of  a  long  compound,  minor  differ- 
ences in  running  may  be  adjusted  by  shifting 
J,  the  end  of  the  one-chord  (see  rule,  page 
56). 

This  should  be  done  by  first  running  out  the  full 
curve  JM,  and  before  attempting  to  put  in  the 
final  spiral. 

In  some  cases  it  will  be  necessary  to  shift  the 
original  P.  C.  C.  before  room  can  be  made  for  end 
spirals. 

In  making  any  or  all  of  these  shifts,  the  nature 
of  the  ground  should  be  kept  in  mind,  in  order  to 
gain  the  advantages  of  a  general  revision  of  the 
line.  For  this  purpose,  a  large-scale  special  plat 
is  often  of  use. 

Fig.  5  indicates  the  process  when  the  curve  is 
to  be  run  in  from  the  lighter  end. 

Here  angle  FGN  =  b. 


THE  SIX-CHORD  SPIRAL 


21 


Then  FG,  parallel  and  equal  to  ED,  =  -£-.  (15) 

cos  b 


(16) 


FN  =  p  tan  &, 

EH  =  ED  +  Pl  =  LM 

Angle  TML  =  c. 
Then 


cos  6 


22  THE  SIX-CHORD  SPIRAL 


TM  =  LM  X  cos  c  =    -£-  +  p     cos  c.      (17) 
\cos  6       r  V 

TU  =  P,  the  required  offset  =  TM  +  MU. 
Hence  the  shift  required  is 


and  the  necessary  pull  back  is 

P  -      —  ^       +  Pi)    COS   C 


sin  (6  +  c)  (19) 

FN  +  TFAT  =  p  tan  6  +  TFAT  (20) 

The  rest  of  the  process  is  the  same  as  in  the  pre- 
ceding case  after  GV  has  been  obtained. 

THE  LENGTH  OF  SPIRALS. 

There  is  no  definite  rule  for  determining  the 
length  of  spirals.  This  depends  on  both  speed 
and  elevation. 

The  rate  on  which  the  given  elevation  is  to  be 
obtained  is  also  important. 

Some  rules  for  spiral  length  are  based  on  a  uni- 
form rate  of  elevation  grade,  such  as  1  in  300,  1  in 
400,  etc. 

The  rational  rule  for  varying  speeds  is  that  the 
same  amount  of  super-elevation  should  be  attained 
in  the  same  time. 

This  may  be  called  the  "time  approach." 

It  follows  that  curves  of  the  same  degree,  oper- 


THE  SIX-CHORD  SPIRAL 


23 


ated  under  different  speed  conditions,  should  have 
spiral  lengths  proportional  to  the  cubes  of  the 
speeds  used. 

The  following  tables  indicate  the  relations  be- 
tween spirals,  and  the  data  used  to  determine 
their  lengths.  Speeds  are  in  miles  per  hour. 
Distances  and  elevations  are  in  feet. 

TABLE  II. 


fl 

J 

*c3 

sq 

i 

.i-a 

f  Jo 

"3 

I 

CO 

O_t- 

CO.tj 

J£  ii  it 

1 

*- 

| 

a 
.2 

•*, 

<—  o. 
oco 

vii  a 
oco 

£> 

|| 

1 

0> 

5*8 

Mo 

5"H 

|j-i-^ 

|l 

|£ 

|1 

5 
w 

j 

J6 

'cd      O 

rt     X 

1 

0-30 

100. 

.5 

3.5 

400.0 

600.0 

1  in  1200 

2 

1- 

70.7 

.5 

3.5 

282.8 

424.2 

1  in    848 

3 

1-30 

57.7 

.5 

3.5 

230.8 

346.2 

1  in    692 

4 

2- 

50.0 

.5 

3.5 

200.0 

300.0 

1  in    600 

5 

2-30 

44.7 

.5 

3.5 

178.8 

268.2 

in    536 

6 

3- 

40.8 

.5 

3.5 

163.2 

244.8 

in    490 

7 

3-30 

37.7 

.5 

3.5 

151.0 

226.5 

in    452 

8 

4- 

35.4 

.5 

3.5 

141.6 

212.4 

in    425 

9 

4-30 

33.3 

.5 

3.5 

133.2 

199.8 

in    400 

10 

5- 

31.6 

.5 

3.5 

126.4 

189.6 

in    379 

In  the  above  table,  the  maximum  safe  speeds 
are,  for  convenience,  taken  as  the  reciprocals  of 
the  square  roots  of  the  degrees  of  main  curve 
X  100;  also, 

Length  of  one-chord  spiral  =  max.  speed  X  4; 

Length  of  six-chord  spiral  =  max.  speed  X  6. 

Note  from  Table  II  that  for  curves  operated 
under  the  same  conditions  of  safe  speed  and  with 


24 


THE  SIX-CHORD  SPIRAL 


time  approaches,  the  offsets  p  and  the  elevation 
are  constant. 

The  following  convenient  rules  for  lengths  of 
spirals  are  also  indicated  by  the  table: 

Rule  1.  —  Length  of  six-chord  equals  speed  in 
miles  per  hour  multiplied  by  elevation  in  inches. 

Here  maximum  p  =  3.5  ft. 

Rule  2.  —  If  somewhat  longer  spirals  be  desired, 
then  length  of  one-chord  spiral  equals  speed  in 
miles  per  hour  multiplied  by  elevation  in  tenths  of 
feet. 

Here  maximum  p  =  5.45  ft. 

Other  rules,  yielding  longer  or  shorter  spirals  as 
desired,  may  be  formed  on  the  same  plan. 

The  following  table  of  elevations  explains  itself. 
The  elevations  are  in  decimals  of  a  foot,  and  the 
speeds  in  miles  per  hour  are  given  at  the  heads  of 
the  columns. 

TABLE  III. 


Degree  of 
Main 

31.6 

33.3 

35.4 

37.7 

40.8 

44.7 

50.0 

57.7 

70.7 

100.0 

Curve 

1 

.05 

.06 

.06 

.07 

.08 

.10 

.13 

.17 

.25 

.50 

2 

.10 

.11 

.13 

.14 

.17 

.20 

.25 

.34 

.50 

3 

.15? 

'.17 

.19 

.21 

.25 

.30 

.38 

.50 

4 

.20 

.22 

.25 

.29 

.33 

.40 

.50 

5 

.25 

.28 

.31 

.36 

.42 

.50 

6 

.30 

.33 

.38 

.43 

.50 

7 

.35 

.39 

.44 

.50 

8 

.40 

.44 

.50 

9 

.45 

.50 

10 

.50 

THE  SIX-CHORD  SPIRAL  25 

Now,  finding  a  5°  curve  which  is  elevated  .36  ft. 
and  giving  satisfaction  as  to  rail  wear,  comfort, 
etc.,  a  glance  at  Table  III  shows  that  it  belongs  to 
the  7°  maximum  series,  having  a  speed  of  37.7 
miles  per  hour. 

The  length  of  spiral  required  would  be  (adopting 
Rule  1  under  Table  II),  .36  X  12  =  4.32  ins.,  and 
4.32  X  37.7  ==  162.9  ft.  for  the  length  of  a  six- 
chord  spiral;  and  162.9  X  f  =  108.6  ft.,  the  cor- 
responding one-chord  spiral. 

If  Rule  2  be  adopted,  then  10  X  .36  X  37.7  = 
135.72  =  length  of  one-chord,  and  135.72  X  1.5 
=  203.58  =  length  of  six-chord. 

It  may  sometimes  be  advisable  to  use  longer 
easements  on  certain  curves,  so  that,  if  the  speed 
limit  be  increased,  the  elevation  only  need  be 
changed,  the  alinement  remaining  fixed. 

For  construction  purposes  it  is  necessary  to 
divide  the  line  into  speed  sections  of  suitable 
length,  treating  each  section  by  itself. 

A  speed  section  may  sometimes  be  as  short  as  a 
single  sharp  curve,  or  even  the  sharp  member  of  a 
compound  curve. 

THE  LENGTH  OF  SPIRALS  JOINING  COMPOUND 
CURVES. 

This  should  obviously  be  sufficient  to  gain  the 
proper  difference  of  elevation  between  the  two 
curves,  or  what  is  the  same  thing,  the  length  for  a 
spiral  from  tangent  to  a  curve  whose  degree  is  the 


26  THE  SIX-CHORD  SPIRAL 

difference  between  the  two  members  of  the  com- 
pound; for  example: 

A  5°  curve  compounds  with  a  3°;  required  the 
length  of  one-chord  connection,  using  Rule  2. 

5°  -  3°  =  2°.  Then,  assuming  speed  at  40.8 
miles,  column  6;  Table  III,  gives  elevation  for  a 
2°  =  .17  ft. 

Then  1.7  X  40.8  =  69.4  =  length  of  one-chord 
spiral.  Length  of  six-chord  =  69.4  X  1.5  =  104.1  ft. 

To  RUN  IN  THE  SIX-CHORD  SPIRAL  BY 

DEFLECTIONS. 

The  degrees  of  curvature  of  the  six  arcs  of  the 
spiral  are: 

D    2D   3D   4D    5D       .  6Z>.  7D 

7'  T'T'  7->Tandy;y==A 

being  the  degree  of  main  curve  (see  Fig.  2). 
The  angle  of  crossing  of  the  six-chord  and  one- 

D  X  C 

chord  at  S3}  or  H3,  =  ?  when  both  D  and  the 

crossing  angle  are  expressed  in  degrees  and 
decimals,  and  C  equals  the  length  of  the  single 
chords  in  feet. 

D  X  L 

The  total  angle  of  the  six-chord  =  — 

.200 

TABLE  IV. 

DEFLECTION  COEFFICIENTS  AND  THEIR  LOGARITHMS  FOR 
SIX-CHORD  SPIRAL. 

These  coefficients  multiplied  by  (C  X  -D) ,  where  C  equals 
chord  length  in  feet,  and  D  equals  degree  of  main  curve  in 


THE  SIX-CHORD  SPIRAL 


27 


degrees,  give  deflections  from  tangent  at  transit  in  minutes 
and  decimals.    Add  the  logarithms  to  log  (C  X  -D). 

S7  is  on  main  curve,  one  chord  length  beyond  S6,  and  is 
given  to  provide  an  alternative  set-up  when  S6  falls  on  bad 
ground. 

The  transit  being  over  any  point  in  the  first  vertical  column, 
the  deflection  coefficients  are  read  from  this  transit  point 
horizontally. 

TABLE  IV. 


Transit 
over 

P.S. 

s, 

S2 

S3 

S4 

S5 

S6 

87 

P.S.  coef.  .  . 
log 

.0429 
8.63202 

.1071 
9.02996 

.2000 
9.30103 

.3214 
9.50708 

.4714 
9.67342 

.6500 
9.81291 

.8571 
9.93305 

Si  coef.  .  . 
log 

.0429 
8.63202 

.0857 
8.93305 

.1929 
9.28524 

.3286 
9.51663 

.4929 
9.69272 

.6857 
9.83614 

.9071 
9.95768 

S2  coef.  .  . 
log 

.1500 
9  .  17609 

.0857 
8.93305 

.1286 
9.10914 

.2786 
9.44494 

.4571 
9  .  66005 

.6643 
9.82236 

.9000 
9.95424 

S3  coef.  .  . 
log 

.3143 
9.49733 

.2357 
9.37239 

.1286 
9.10914 

.1714 
9.23408 

.3643 
9.56144 

.5857 
9.76769 

.8357 
9.92206 

84  coef.  .  . 
log 

.5357 
9.72893 

.4429 
9.64626 

.3214 
9.50708 

.1714 
9.23408 

.2143 
9.33099 

.4500 
9.65321 

.7143 

9.85387 

S5  coef.  .  . 
log 

.8143 
9.91078 

.7071 
9.84951 

.5714 
9.75696 

.4071 
9.60975 

.2143 
9  .  33099 

.2571 
9.41017 

.5357 
9.72893 

S6  coef.  .  . 
log 

1  .  1500 
0.06070 

1.0286 
0.01223 

.8786 
9.94378 

.7000 
9.84510 

.4929 
9.69272 

.2571 
9.41017 

.3000 
9.47712 

S7  coef.  .  . 
log 

1.5429 
0.18833 

1.4071 
0.14834 

1.2429 
0.09442 

1.0500 
0.02119 

.8286 
9.91833 

.5786 
9.76236 

.3000 
9.47712 

Total  1  coef. 
angle  i  log 

P.S.  to 

0.0857 
8.93305 

0.2571 
9.41017 

0.5143 
9.71120 

0.8571 
9.93305 

1.2857 
0.10914 

1.8000 
0.25527 

2.4000 
0.38021 

The  total  angle  of  the  six-chord  spiral  in  min- 
utes =  C  X  D  X  1.8. 

The  degrees  of  curvature  of  the  six-chord  spiral 

6D 

—  . 

p  =  length  of  spiral  X  sine  of  deflection  angle 
P.  S.  to  S3. 
See  also  formula  (4),  page  9. 


, 

arcs  are  —  to 


28 


THE  SIX-CHORD  SPIRAL 


Example.  —  Take  a  14°  curve  having  a  spiral 
approach  of  six  chords,  each  25  ft.  long  or  150  ft. 
in  all,  to  calculate  the  deflections. 

Here  C  X  D  =  25  X  14  =  350.  (log  =  2.54407). 

Then  from  Table  IV,  instrument  on  P.  S., 


8.63202  9.02996 
2.54407  2.54407 
1.17609  1.57403 


15' 
0°15' 


37.5' 
0°  37J' 


9.30103 

2.54407 

1.84510 

70' 


9.50708 
2.54407 


112.5' 
1°  52  J' 


9.67342 
2.54407 


165' 

2°  45' 


9.81291 
2.54407 


2.05115    2.21749    2.35698 


227.5' 

3°  47  J' 


With  instrument  at  $6,  to  turn  tangent  to  the 
six-chord  and  main  curve  at  S9: 

Sight  on  P.  S.  with  vernier  set  at  (see  Table  IV) 
1.15  X  C  X  D  =  1.15  X  350  =  402J'  -  6°  42J', 
and  then  turn  vernier  to  zero.  Or,  sighting  on 
S3,  0.7  X  350  =  245'  =  4°05',  which  is  to  be 
turned  off  at  S9  to  obtain  tangent. 

These  computations  may  be  made  by  logarithms, 
as  before. 

For  instrument  at  S3  the  crossing  angle  be- 
tween the  spiral  and  the  7°  curve  (one-chord 


THE  SIX-CHORD  SPIRAL  29 


spiral)  will  be  C  X  D  +  700  =  350  -*-  700  =  0.5° 
=  0°  30',  and  from  this  one  may  pass  from  one 
curve  to  the  other. 
The  total  angle  of  the  six-chord  is 

D  X  L  -i-  200  =  14  X  150  4-  200  =  10°  3t)'. 

To  calculate  the  deflections  for  a  six-chord  spiral 
joining  two  members  of  a  compound  curve  (see  ex- 
ample under  Fig.  3) : 

First  calculate  the  deflections  by  Table  IV  for 
150  ft.  of  six-chord  spiral  joining  a  tangent  with  a 
10°  40'  -  6°  *=  4°  40'  main  curve. 

Then  to  each  deflection  thus  found  add  that  of 
a  6°  curve  for  the  length  of  sight  taken. 

Thus,  from  P.  S.  to  S,  add  45' ;  from  S3  to  SQ  add 
2°  15'. 

If  so  desired,  necessary  tabulations  may  be  pre- 
pared in  advance,  giving  once  for  all  the  deflections 
required  for  the  general  run  of  curves  in  use,  pre- 
cisely as  is  customary  with  all  table  spirals. 

THE  TRACK  PARABOLA. 

Table  V  may  be  used  in  offsetting  from  the  one- 
chord  spiral  to  the  track  parabola. 

Tables  I  and  V  are  on  the  same  six-chord  base 
and  may  be  similarly  used. 

It  will  be  noticed  that  the  differences  between 
the  corresponding  offsets  in  Tables  I  and  V  are, 
for  any  usual  value  of  p,  too  small  to  be  note- 
worthy. 


30 


THE  SIX-CHORD  SPIRAL 


In  actual  service,  the  parabola  has  no  advantage 
whatever  over  the  polychord  spiral,  and  a  choice 
between  them  should  be  governed  by  their  rela- 
tive adaptability  to  field  and  office  use. 

The  offsets  in  Table  V  are  to  be  measured  in- 
ward from  the  main  tangent  half  of  spiral,  and 
outward  from  the  main  curve  half. 

Note  that  the  offsets  at  P.  S.  are  insignificant. 

For  p  =  10  ft.  they  are  0.01  ft. 

TABLE  v. 

TABLE  OF  INTERMEDIATE  OFFSETS  TO  TRACK  PARABOLA 
FROM  MAIN  TANGENT  AND  MAIN  CURVE  WITH  ONE- 
CHORD  APPROACH. 


! 

X^ 

js  8 

o<£ 

1 

JS 

! 

,c 

"So 
1 

ii 

1 
£ 

J3 

* 

^ 

«, 

X£ 

X  o> 

O  **-! 

1 

jj> 

1 

Tenths  of  chord 

Coefficients  whi 
give  offsets  in 

Tenths  of  chord 

Differences  for  < 
hundredth  of 
chord  length 

I  Tenths  of  chord 

Coefficients  whi 
give  offsets  in 

Tenths  of  chord 

Differences  for  i 
hundredth  of 
chord  length 

Tenths  of  chord 

Coefficients  whii 
give  offsets  in 

Tenths  of  chord 

Differences  for  < 
hundredth  of 
chord  length 

P.S. 

.001 

S6 

Si 

.038 

S5 

S2 

.055 

S4 

.0001 

.0007 

.0003 

1 

.002 

9 

1 

.045 

9 

1 

.052 

9 

.0001 

.0006 

.0004 

2 

.003 

8 

2 

.051 

8 

2 

.048 

8 

.0002 

.0004 

.0005 

3 

.005 

7 

3 

.055 

7 

3 

.043 

7 

.0003 

.0003 

.0005 

4 

.008 

6 

4 

.058 

6 

4 

.038 

6 

.0003 

.0002 

.0006 

5 

.011 

5 

5 

.060 

5 

5 

.032 

5 

.0004 

.0001 

• 

.0006 

6 

.015 

4 

6 

.061 

4 

6 

.026 

4 

.0004 

.0000 

.0006 

7 

.019 

3 

7 

.061 

3 

7 

.020 

3 

.0005 

.0001 

.0006 

8 

.024 

2 

8 

.060 

2 

8 

.014 

2 

.0007 

.0002 

.0007 

9 

.031 

1 

9 

.058 

1 

9 

.007 

1 

.0007 

.0003 

.0007 

s, 

.038 

S5 

S2 

.055 

S4 

sa 

.000 

S3 

THE  SIX-CHORD  SPIRAL  31 

RELATIVE   LENGTHS   AND  TOTAL  ANGLES   OF 
SPIRALS,  p  AND  RM  CONSTANT  (see  Fig.  1) : 
Let    L!  =  length  or  total  angle  of  one-chord  spiral. 
L6  =  length  or  total  angle  of  six-chord  spiral. 
LP  =  length  or  total  angle  of  track  parabola. 
Then  L6  =  1.5  L,       L,  =  f  L6         L,  =  .577  LP 
LP  =  1.733  Ll    LP  =1.155  L6   L6  =  .866  LP 
Example.  —  Given  RM  =  1432.5  =  4°  curve, 

p    -  4.65; 
The  total  angle  of  a  one-chord  will  be  [  (3),  page 

8] 

4°  37'  -  4.617° 

The  total  angle  of  a  six-chord  = 

4.617°  X  1.5  =  6.926°  =  6°  55 \' 
The  total  angle  of  track  parabola  = 
4.617°  X  1.733  =  8°  00' 

Length  one-chord  =      4.617  -f-  2     =  230.85  ft. 

Length  six-chord  =  230.85     X  1.5  =  346.28  ft. 

Length    parabola  =  230.85  X  1.733  =  400.00  ft. 

These  lengths  are  bisected  at  S3,  which  is  the 
middle  point  of  all  spirals. 

In    the    foregoing    example,    400  -  230.85  = 
169.15  ft.  is  the  difference,  LP  -  L1  =  .733  Lt. 

Hence,  169.15  H-  2  =  84.58  ft.,  is  the  distance 
to  be  laid  off  along  main  tangent  or  main  curve 
from  the  beginning  or  ending  of  the  one-chord,  in 
order  to  obtain  the  beginning  or  ending  of  the 
track  parabola.  This  may  be  used  in  connection 


32  THE  SIX-CHORD  SPIRAL 

with  Table  V,  when  it  is  desired  to  lay  off  the  track 
parabola. 

The  total  angles  of  the  spirals  will  be  divided  at 
the  middle  point  S3  as  follows: 
One-chord  4.617°,  J  -  2.31°  on  tangent  half. 
One-chord  4.617°,  J  -  2.31°  on  main  curve  half. 
Six-chord  6.926°,  f  =  1.98°  on  tangent  half. 
Six-chord  6.926°,  f  =  4.95°  on  main  curve  half. 
Track  parabola  8°,  J  =       2°  on  tangent  half. 
Track  parabola  8°,  f  =       6°  on  main  curve  half. 

In  all  spiral  running  it  is  important  to  keep  a 
watch  on  the  total  angles  of  the  various  parts,  so 
that  the  grand  total,  from  tangent  to  tangent,  will 
check  with  the  intersection  angle  of  the  whole 
curve. 

DEMONSTRATION  OF  THE  SIX-CHORD  SPIRAL. 

In  this  spiral  (Fig.  6),  if  the  total  angle  of  the 
first  arc,  P.  S.  to  Sly  be  taken  as  2,  that  of  the  second, 
Sfiz,  will  be  4,  the  third,  S2S3,  6,  and  so  on,  S6S7 
being  14.  $6*S7  coincides  with  the  main  curve,  the 
end  of  spiral  being  at  $6,  all  chords  being  of  the 
same  length. 

Hence  the  angles  which  the  spiral  makes  with 
the  outer  tangent  will  be  at  SltS2,  etc.,  2,  6,  12,  20, 
30,  42,  and  56,  the  angle  42,  at  S9,  being  the  total 
angle  of  the  spiral. 

The  angle  which  each  chord  of  the  spiral, 
P,  S.  Si,  S1S2,  etc.,  makes  with  the  outer  tangent 


THE  SIX-CHORD  SPIRAL 


33 


34  THE  SIX-CHORD  SPIRAL 

will  be  the  total  angle  to  the  end  of  that  chord 
less  the  deflection  angle  of  the  last  arc. 

From  P.  S.  to  Sl  it  equals  2-1  =  1;  S&, 
6  —  2=4,  and  so  on,  or  as  the  squares  of  the 
natural  numbers. 

Since  the  sines  of  small  angles  are  proportional 
to  the  angles,  the  ordinates  from  S1}  S2,  etc.,  will 
be  as  the  sums  of  these  squares,  or  as  1,  5,  14,  30, 
55,  91,  and  140,  as  marked  on  the  figure.  AB  = 
140  -  91  =  49. 

Since  the  total  angle  of  the  spiral  to  $6  is  repre- 
sented by  42,  and  to  S7  by  56,  the  angle  S7OS& 
equals  56  -  42  =  14,  both  on  main  curve  and 
spiral.  Now,  as  14  is  one-fourth  of  56,  continuing 
the  main  curve  back  to  D  through  SQ  and  H5  will 
make  the  tangent  at  D  parallel  to  the  outer  tan- 
gent. The  angles  K,  L,  and  M  each  being  equal 
to  SQOS7,  0  will  be  at  right  angles  to  DF  at  D. 

Assuming  that  the  versed  sines  of  small  angles 
are  proportional  to  the  squares  of  those  angles,  we 
have  AD:  BD::4?:&  =  16:9. 

Hence,  AD  -  BD:BD::IQ  -  9:9.  But  AD 
-  BD  =  49,  consequently, 

49:£D::7:9.    :.BD  =  63,  and^D  =  112. 

Take  H5  on  the  main  curve,  so  that  S6H5  sub- 
tends the  angle  M  and  equals  SQS7 ',  then  AD :  CD : : 
42:  22,  and  CD  =  28  =  }  AD.    Also  ED  =  140  - 
112  =  28,  and  DS3  =  14  =  S3E. 

Now  a  circle  of  twice  the  radius  OS7,  tangent  at 
#§,  will  in  4  chord-lengths  have  a  versed  sine  = 


THE  SIX-CHORD  SPIRAL  35 

CE  or  28  X  2  =  56,  and  be  tangent  to  the  outer 
tangent  at  H^ 

Taking  the  ordinates  to  this  circle  proportional 
to  the  square  of  the  number  of  chords,  it  will  pass 
through  S3,  and  the  ordinates  to  it  will  be  at  H1  = 
zero,  at  H2  =  f|  =  3},  #4  =  T9e  X  56  =  31}. 
Hence  H&  =  1,  H2S2  =  5  -  3}  =  1},  H,S4  = 
31}  -  30  =  1},  and  H5S5  =  56  -  55  =  1,  or,  in 
terms  of  the  main  offset,  p  =  28,  H^  =  H5S5  = 
.036  p,  and  H2S2  =  H,S4  =  .054  p. 


COMPARATIVE  TABULATIONS  SHOWING  THE  RELA- 

TION BETWEEN  THE  SIX-CHORD  SPIRAL  AND 

TERMINAL  CURVE  WHEN  EACH  is  EXACTLY  AND 

INDEPENDENTLY  CALCULATED. 

The  following  tables  give  the  coordinates  of  the 

H  points  and  the  S  points,  by  corresponding  pairs, 

on  three  typical  spirals.     In  each  case  the  spiral 

and  terminal  curve  are  taken  to  run  in  a  north- 

westerly direction  from  a  main  tangent  running 

due  north.     For  convenience  in  taking  out  sines 

and  cosines  from  table  direct,  each  chord  is  100 

feet  long.     Other  spirals  having  the  same  total 

angle  may  be  formed  by  multiplying  the  tabular 

quantities   by   the   selected   chord  length  -=-  100. 

In  this  case  the  degrees  of  the  main  and  terminal 

curves  will  equal  the  degrees  given  in  the  tables 

chord 


luU 
curve  at  each  point  is  also  given. 


36 


THE  SIX-CHORD  SPIRAL 


The  differences  between  corresponding  pairs  of 
points  $!  and  Hlf  S2  and  H2,  etc.,  are  taken  from 
each  H  point  as  an  origin  or  zero;  thus  the  differ- 
ence between  H2  and  S2  (in  Table  VI)  of  W.  .437 
and  S.  .002  means  that  S2  lies  west  and  south  of 
H2,  .437  and  .002  feet  respectively. 

TABLE  VI. 

COORDINATES  FOR  600  FEET  OF  SIX-CHORD-SPIRAL  APPROACH 
TO  2°  20'  MAIN  CURVE,  AND  ALSO  FOR  400  FEET  OF  1°  10' 
TERMINAL  CURVE  JOINING  SAME  TANGENT  AND  CURVE. 

RM  =  2455.7  ft.,  p  =  8.15  ft.,  spiral  angle  =  7°,  terminal 
angle  =  4°  40'.  Spiral  angle  of  corresponding  track  para- 
bola =  8°  05'. 

p  X  .036  =  .293  ft.;"  p  X  -054  =  .44  ft. 


0^ 

£ 

~ 

"84. 

£ 

d 

c  « 

3 

"S 

"c 

1 

o 

.-.  W) 

d 

** 

'S 

.£  W) 

rf 

•*-> 

PH 

JJ 

a 

3 

£ 

IP 

1 

a 

H, 

N 

0.000 

100.000 

H4 

N  30  30'W 

9.160 

399.818 

s, 

N  20'W 

.291 

100.000 

N  3°  20'W 

8.726 

399.851 

W.291 

0.000 

E.434 

N.033 

H, 

N  10  10'W 

1.018 

199.995 

H5 

N  40  40'W 

16.281 

499.564 

So 

N  1°  W 

1.455 

199.993 

S. 

N  50  00'  W 

15.992 

499.587 

W.437 

S.002 

E.289 

N.023 

HS 

N  20  20'W 

4.072 

299.948 

Ho 

N  7o  W 

26.445 

599.046 

S3 

N  20  W 

4.073 

299.959 

Se 

N  7°  W 

26.445 

599.039 

W.001 

N.011 

0.000 

S.007 

TABLE   VII. 

COORDINATES  FOR  600  FEET  OF  SIX-CHORD  SPIRAL  APPROACH 
TO  4°  40'  MAIN  CURVE,  AND  ALSO  FOR  400  FEET  OF  2°  20' 
TERMINAL  CURVE  JOINING  SAME  TANGENT  AND  CURVE. 

RM  =  1228.1  ft.,  p  =  16.26  ft.,  spiral  angle  =  14°,  terminal 
angle  =  9°  20'.  Spiral  angle  of  corresponding  track  para- 
bola =  16°  10'. 

p  X  .036  =  .585  ft.;  p  X  .054  =  .878  ft. 


THE  SIX-CHORD  SPIRAL 


37 


O  +* 

"S-a 

£ 

«a 

3 
** 

3 

a 

M§ 

2 

| 

2 

|| 

a 

•5 

£ 

11 

1 

1 

rH 

IH 

a 

3 

£H 

& 

5 

Hi 

N    OOW 

0.000 

100.000 

H4 

N  70W 

18.305 

399.274 

Si 

N  40'W 

.582 

99.998 

S4 

N  60  40'W 

17.438 

399.401 

W.582 

S.002 

E.867 

N.127 

H2 

N  20  20'W 

2.036 

199.979 

H5 

N  90  20'W 

32.510 

498.260 

S2 

N  20W 

2.909 

199.971 

S5 

N  10°  OO'W 

31.931 

498.345 

W.873 

S.008 

E.579 

N.085 

H3 

N  40  40'W 

8.141 

299.792 

Ho 

N  140  OO'W 

52.732 

596.194 

S3 

N  40W 

8  143 

299.834 

N  140  OO'W 

52.722 

596.160 

W.002 

N042 

E.010 

S.034 

TABLE  VIII. 

COORDINATES  FOR  600  FEET  OF  SIX-CHORD  SPIRAL  APPROACH 
TO  7°  MAIN  CURVE,   AND   ALSO  FOR  400  FEET  OF  3°  30' 
TERMINAL  CURVE  JOINING  SAME  TANGENT  AND  CURVE. 
RM  =  819.9,  p  =  24.35   ft.,  spiral   angle  =  21°,  terminal 
angle  =  14°.      Spiral   angle    of    corresponding   track    para- 
bola =  24°  15'. 

p  X  .036  =  .877  ft.;  p  X  .054  =  1.315  ft. 


a 

P 

§ 

| 

g 

°"c 

1 

1 

1 

9 

I 

1 

1 

V 

I 

3 

HI 

N  0°  W 

0.000 

100.000 

H4 

N  10^30'W 

27.416 

398.369 

Si 

N  P  W 

.873 

99.996 

S4 

N  10°W 

26.126 

398.654 

W.873 

S.004 

El.  290 

N.285 

H2 

N  3030'  W 

3.054 

199.953 

H5 

N  140W 

48.634 

496.092 

S2 

N  30W 

4.363 

199.935 

S5 

N  150W 

47.770 

496.284 

W  1.309 

S.018 

E.864 

N.192 

H3 

N  7°W 

12.204 

299.533 

HO 

N  2FW 

78.705 

591.464 

S3 

N  60W 

12.209 

299.627 

SK 

N  2PW 

78.672 

591.390 

W.005 

N.094 

E.033 

S.074 

An  inspection  of  these  tables  shows : 
1st.   That  in  all  three  cases  the  six-chord  spiral 
practically  passes  through  H3.    In  Table  VIII  (an 
extreme  case  of  high  values  for  p  and  spiral  angle) 


38  THE  SIX-CHORD  SPIRAL 

S3  is  W.  .005  and  N.  .094  of  H3,  and  the  tangent  to 
the  curve  at  S3  bears  N.  6°  W.  Tracing  the  six- 
chord  south  for  .094  of  latitude  would  reduce  its 
departure  .094  X  tangent  6°  (.105)  -  .010,  which 
would  cause  the  six-chord  to  pass  .010  —  .005  = 
.005  feet  due  east  of  H3. 

In  Table  VII  S3  would  fall  .001  feet  due  east 
of  H3. 

2d.  That  in  all  three  cases,  the  six-chord  spiral 
(continued)  practically  passes  through  H6,  which 
is  on  the  main  curve  one  chord  length  beyond  H5. 

Thus,  in  Table  VIII,  S6  lies  E.  .033  and  S.  .074 
feet  of  H6,  and  the  tangent  to  the  curve  bears 
N.  21°  W.  A  continuation  along  this  tangent  for 
N.  .074  feet  would  make  a  westing  of  .074  X  tan- 
gent 21°  (.38)  =  .028  feet,  and  the  six-chord 
would  pass  .033  -  .028  =  .005  feet  due  east 
of#a. 

It  is  to  be  noted  that  continuing  the  six-chord 
.074  north  would  lengthen  it  along  the  7°  curve 
.074  •*-  cos  21°  (.93)  =  .08  feet,  thus  increasing 
the  total  angle  to  the  point  abreast  of  HQ  by  7°  X 
.6  X  .08  =  J  minute,,  which,  in  this  extreme  case, 
would  be  the  error  in  total  angle. 

Note  also  that  the  coordinates  of  S6  divided 
one  by  the  other  give  78.672  -j-  591.39  =  .13303 
=  tangent  7°  34f.  Now  the  table  of  deflections 
for  a  six-chord  previously  given  shows  a  deflection 
from  P.  S.  to  &  of  C  X  D  X  0.65  -  100  X  7°  X 


THE  SIX-CHORD  SPIRAL  39 

0.65  =  455'  =  7°  35'.    Here  also  is  an  error  of  J 
minute. 

A  tabulation  similar  to  VIII  but  reversed,  i.e., 
starting  from  SQ  and  running  back  to  the  P.  S., 
gives  for  the  quotient  of  the  coordinates  of  P.  S., 
138.492  -  580.303  =  .23865  -  tangent  13°  25}'. 
By  table  of  deflections  this  angle  is  C  X  D  X 
1.15  =  100  X  7  X  1.15  =  13°  25',  or  again  an  error 
of  }  minute. 

Similar  computations  will  show  that  the  errors 
for  all  intermediate  deflections  are  insignificant. 

The  same  treatment  of  Tables  VI  and  VII 
will  show  no  material  error  whatever,  that  in 
Table  VII  from  P.  S.  to  S6,  or  S6  to  P.  S.,  being 
only  i1-^  of  one  minute. 

3d.  A  comparison  of  the  actual  offsets  between 
the  two  curves  at  Hlt  H2,  H4,  and  H5  is  best  made 
by  platting  the  coordinates  of  the  S  points  with 
reference  to  their  corresponding  H  points,  on  a 
scale  of  ten  inches  to  the  foot,  and  drawing  the 
tangents  through  each  pair  of  points  from  the 
bearings  given  in  the  tables.  By  this  it  will  be 
found  that  in  every  case  (measuring  at  right  angles 
to  the  H  line)  the  coefficients  .036  and  0.54  mul- 
tiplied by  p  will  give  the  correct  distance  between 
the  two  curves,  almost  exactly. 

From  the  foregoing  the  conclusion  is  drawn 
that,  even  for  unusually  large  values  of  p  and  the 
spiral  angle,  the  method  of  offsets  from  the  ter- 


40  THE  SIX-CHORD  SPIRAL 

niinal  curve  to  the  six-chord  spiral  is  practically 
exact,  and  that  the  methods  of  offsets  and  deflec- 
tions are  interchangeable,  i.e.,  one  method  will 
duplicate  the  other  theoretically  much  closer  than 
either  can  be  made  to  duplicate  itself  on  the 
ground,  with  the  customary  appliances  and 
methods. 

COMPARISON  OF  SPIRALS  AND  SUMMARY. 

The  railroad  spiral  provides  for  a  gradual  change 
from  the  position  of  car  and  trucks  on  a  tangent 
to  that  assumed  by  them  on  a  curve. 

This  change  is  effected  by  an  intermediate  curve 
having  an  average  curvature  usually  one-half  that 
of  the  main  curve. 

Figure  1  shows  the  general  problem.  Here 
FJN  is  the  main  curve  with  center  at  C",  and 
HGE  the  main  tangent.  The  main  curve  has  been 
moved  inward  a  distance  BN  from  its  original 
position.  This  shift  is  necessary  to  allow  room 
for  the  insertion  of  the  lighter  intermediate  curve. 
The  new  main  curve  merges  into  the  shift  tangent 
(which  is  parallel  to  the  main  tangent)  at  F. 

The  simplest  form  of  spiral  is  that  shown  by  the 
dotted  curve  HKJ,  which  has  twice  the  radius  or 
half  the  degree  of  the  main  curve.  This  is  called 
the  terminal  curve  or  one-chord  spiral.  The  point 
K,  which  practically  bisects  the  principal  offset, 
FG  =  p,  is  the  middle  point  of  the  length  of  the 
spiral. 


THE  SIX-CHORD  SPIRAL  41 

H  is  the  P.  C.  and  J  the  P.  C.  C.  of  the  one-chord. 

If  two  curves  be  used  in  passing  from  tangent  to 
curve,  the  degree  of  the  first  will  be  from  tangent 
to  K  =  J  degree  of  main  curve,  and  of  the  second 
from  K  to  main  curve  =  f  of  the  same.  As  before, 
K  is  the  half-way  point  of  the  spiral.  This  con- 
stitutes a  two-chord  spiral. 

Calling  the  total  length  of  the  one-chord  unity, 
that  of  the  two-chord  will  be  1.225.  Hence  the 
latter  starts  from  the  tangent  to  the  left  of  H,  and 
passing  through  K  merges  into  the  main  curve 
between  J  and  N.  It  thus  lies  inside  the  one- 
chord  from  tangent  to  K,  and  outside  from  K  to 
main  curve,  the  two  spirals  crossing  each  other 
at  K. 

This  condition  is  indicated  by  the  dotted  spiral 
in  Fig.  2,  where  Hlf  H3,  and  H5  are  points  on  the 
one-chord  and  respectively  correspond  with  H,  K, 
and  J  of  Fig.  1. 

Spirals  having  any  number  of  chords  (N)  are 
so  taken  that  the  degree  of  curve  of  the  first  arc 
=  degree  of  main  curve  H-  (N  +  1),  that  of  the 
second  twice,  of  the  third  thrice  that  of  the  first 
arc,  and  so  on,  the  (N  +  1)  arc  coinciding  with  the 
main  curve. 

The  lengths  of  spirals  for  fixed  values  of  p  and 
main  curve  increase  with  the  number  of  chords  or 
arcs  used;  that  is,  they  start  further  back  on  the 
tangent,  and,  passing  through  the  common  point 


42  THE  SIX-CHORD  SPIRAL 

K  (where  they  are  bisected),  reach  further  around 
the  main  curve  toward  N,  Fig.  1,  before  merging 
into  it. 

The  greater  the  number  of  arcs  in  a  spiral,  the 
greater  the  lateral  deviation  from  the  one-chord 
on  the  inside  of  KH  and  the  outside  of  KJ. 

The  limit  is  reached  when  the  number  of  arcs 
becomes  infinite.  The  spiral  then  increases  uni- 
formly in  curvature  from  start  to  finish  and  with- 
out pause.  This  constitutes  the  usual  track 
parabola  whose  length  is  1.733  that  of  the  one- 
chord. 

The  curvature  of  all  spirals  increases  at  the 
same  rate  from  K  toward  the  main  curve  as  it 
decreases  from  K  toward  the  main  tangent. 

Hence,  with  degree  of  main  curve  and  p  fixed,  the 
total  angle  of  a  spiral  is  proportional  to  its  length. 

The  total  angles  and  lengths  of  various  spirals 
are  given  in  the  following  table,  those  of  the  one- 
chord  being  unity: 

1-chord  =  1.000  10-chord  =  1.581 

2-chord  =  1.225  11-chord  =  1.593 

3-chord  =  1.342  12-chord  =  1.604 

4-chord  -  1.414  13-chord  -  1.613 

5-chord  =  1.464  14-chord  =  1.621 

6-chord  =  1.500  15-chord  =  1.628 

7-chord  -  1.528  16-chord  =  1.634 

8-chord  =  1.549  17-chord  -  1.640 

9-chord  =  1.567  parabola  =  1.733 


THE  SIX-CHORD  SPIRAL  43 

Example.  —  Take  RM  =  286.5  ft.  (20°  curve)  and 
p  =  4.35  ft.,  the  total  angle  of  the  one-chord 
being  10°. 

These  conditions  will  be  fitted  by: 

100  feet  of  one-chord,  total  angle  10°. 

150  feet  of  six-chord,  total  angle  15°. 

156.7  feet  of  nine-chord,  total  angle  15°  40'. 

173.3  feet  of  parabola,  total  angle  17°  20'. 

Each  chord  of  the  six-chord  will  be  150  -=-  6  = 
25  ft.,  and  of  the  nine-chord,  156.7  -j-  9  =  17.4  ft. 

The  degrees  of  curve  of  the  six  arcs  of  the  six- 
chord  will  be  -V-,  -VH  -6T°->  ¥>  --?->  and  1f--  The 
seventh  or  (N  +  1)  arc  is  J-f-2-,  which  is  the  20° 
main  curve.  In  this  example,  the  difference  (173.3 
-  150)  divided  by  2  ( =  11.65  ft.)  is  the  amount  the 
parabola  overlaps  the  six-chord  at  each  end. 

The  lateral  variation  of  any  of  these  spirals  from 
the  one-chord  or  from  each  other  is  the  same  at 
equal  distances  from  K  measured  along  the  spiral, 
but  these  offsets  are  to  be  made  inward  from  the 
one-chord  on  the  main  tangent  side  of  K,  and 
outward  on  the  main  curve  side. 

From  this  it  follows  that  the  total  length  of 
track  between  common  points  on  the  main  tan- 
gent and  main  curve  is  the  same  for  fixed  values 
of  RM  and  p,  no  matter  what  spiral  be  used,  so 
that  after  track  has  been  laid  to  a  one-chord  it 
may  be  shifted  to  a  track  parabola  or  any  inter- 
mediate spiral  without  altering  the  expansion. 


44  THE  SIX-CHORD  SPIRAL 

For  any  one  form  of  spiral  with  a  fixed  value  of 
RM,  the  principal  offset  p  varies  as  the  square  of  the 
length  of  the  spiral;  that  is,  doubling  the  length  of 
spiral  increases  p  four  times. 

If  the  distances  along  the  one-chord  from  the 
middle  point  K  or  from  either  end  be  expressed 
in  fractions  of  the  length  of  the  one-chord,  then 
the  offsets  from  the  one-chord  to  any  fixed  form  of 
spiral  at  any  given  point  will  equal  p  X  constant 
coefficient  for  that  point,  regardless  of  the  degree 
of  main  curve  or  length  of  spiral.  Thus,  in  Fig.  2, 
the  offsets  S2H2  or  H4S4,  which  are  at  the  quarter 
points  of  the  one-chord,  will  always  for  a  six-chord 
spiral,  equal  p  X  .054.  From  the  same  quarter 
points  of  the  one-chord  to  the  parabola  the  offsets 
are  always  p  X  .055. 

The  complete  coefficients  for  the  six-chord  and 
track  parabola  are  given  in  Tables  I  and  V;  see 
also  Fig.  2. 

Since  the  length  of  the  six-chord  is  always  1.5 
times  that  of  the  one-chord,  the  quarter  points  of 
the  one-chord  lie  abreast  of  the  sixth  points  of  the 
six-chord.  Both  Tables  I  and  V  give  the  offsets 
at  the  various  points  along  the  six-chord,  from  its 
beginning  at  P.  S.  through  Slt  S2,  etc.,  to  its  end 
at  S6.  This  is  solely  for  convenience  in  setting 
off  and  in  making  comparisons. 

These  coefficients  are  the  offsets  in  feet  when 
p  =  1  ft.  For  any  other  value  of  p,  multiply  by  p. 


THE  SIX-CHORD  SPIRAL  45 

Thus,  when  p  =  10  ft.  (an  unusually  large 
value),  Table  I  shows  that  the  maximum  distance 
from  the  six-chord  to  the  one-chord  is  .059  X  10 
=  .59  ft.  at  1.7  chord  lengths  from  either  the 
beginning  or  end  of  the  six-chord. 

Table  V  shows  that  the  maximum  distance  of 
the  parabola  from  the  one-chord  is  at  1.65  chord 
lengths  from  the  beginning  or  end  of  the  six-chord, 
and  equals  (with  p  =  10  ft.)  0.061  X  10  =  .61  ft. 

Comparing  I  and  V  shows  that  the  greatest 
divergence  of  the  parabola  from  the  six-chord 
occurs  at  1.2  chord  lengths  from  the  beginning  or 
ending  of  the  six-chord  and  equals  (.051  —  .048) 
X  10  =  0.03  ft. 

Table  V  also  shows  that  the  offset  from  main 
tangent  and  main  curve  at  the  beginning  and 
ending  of  the  six-chord  (at  P.  S.  and  $6)  equals 
p  X  0.001,  which,  when  p  =  10  ft.,  becomes  .01  ft. 

Hence,  for  easement  purposes,  the  excess  of  length 
of  the  parabola  over  the  six-chord  is  negligible. 


PART  II. 
SPIRALING  OLD  TRACK. 

Spiraling  old  track  consists  mainly  in  com- 
pounding to  make  room  for  the  spirals. 

The  methods  used  for  the  shifts  are  entirely  in- 
dependent of  the  form  of  spiral,  for,  with  fixed 
values  of  p  and  RM,  any  spiral  from  the  one-chord 
to  the  track  parabola  may  be  inserted,  differing 
from  each  other,  of  course,  in  length  and  total 
angle,  according  to  Table  IX,  but  all  giving  prac- 
tically the  same  length  of  line  between  common 
points. 

In  making  room  for  a  six-chord  spiral,  the 
obvious  method  is  to  first  provide  for  a  one-chord, 
remembering  that  the  one-chord  radius  must  be 
double  that  of  the  revised  curve  into  which  it  com- 
pounds, and  not  double  that  of  the  existing  curve, 
unless  the  latter  be  unchanged. 

With  this  condition  imposed,  any  of  the  formulas 
for  three-center  compound  curves  may  be  used 
direct. 

Space-shifts  for  inserting  spirals  are  usually 
made  according  to  one  or  the  other  of  the  following 
assumptions : 

1st.  To  leave  as  much  of  the  original  line  un- 
disturbed as  circumstances  permit. 

46 


THE  SIX-CHORD  SPIRAL  47 

2d.  To  preserve  the  original  length  of  line,  thus 
avoiding  numerous  equations  of  distance. 

In  either  case  the  tangents  are  usually  undis- 
turbed, the  necessary  changes  being  confined  to 
the  curves. 

When  working  on  the  first  assumption,  the 
following  compound-curve  formulas  are  most  useful 
(see  following  example  for  application): 

R          =  R    -  vers/  (21) 

P 
p          =(R0-  RN)  vers  /.       (22) 

Vers/--^-^-.  (23) 

where  RN  =  radius  of  new  main  curve, 

R0  =  radius  of  original  main  curve, 
p     =  principal  offset. 

/  equals  the  angle  cut  out  of  the  R0  -curve  and 
replaced  by  the  RN-curve. 

The  degree  of  the  RN-curve  is  usually  taken  from 
one-tenth  to  one-fifth  greater  than  the  degree  of 

RO- 
The  one-chord  terminal  angle  Tl  is  determined 

from  vers  7\  =^-,  and  either  the  one-chord  or 

six-chord  run  in. 

The  P.  C.  of  the  one-chord,  2RN}  will  be  back 

along  the  main  tangent  a  distance  from  the  original 

P.  C.  =RN  sin  Tl  -  pcoti  /  (24). 


48  THE  SIX-CHORD  SPIRAL 

Example.  —  To  replace  one  end  of  a  4°  curve 
with  enough  4°  30'  to  give  an  offset  p  =  6.62  ft. 

Here  vers  /  =  — — '         =  -^  =  .0416. 
HO  —  HN       159 

Hence,        /  =  16°  35';  and 

16°  35'  of  4°        =  414.6  ft.,  also, 
16°  35'  of  4°  30'  =  368.5  ft. 

Hence,  the  last  414.6  feet  of  the  4°  is  to  be  re- 
placed by  368.5  feet  of  4°  30'  curve. 

Again,  vers  Tl  =  ^-   =  79~^  =  .0052 

T,  =  5°  50.6' 
5°  50.6'  of  4°  30'  =  ^|^  =  129.87  ft., 

which  in  its  turn  is  replaced  by 

129.87  X  2  =  259.74  ft.  of  2°  15' 
one-chord  approach. 

From  the  preceding  formula  this  2°  15'  one- 
chord  will  begin  on  main  tangent  back  from  origi- 
nal P.  C.  a  distance  = 

(1273.6  X  .1018)  -  (6.62  X  6.862)  =  84.2  ft. 

It  is  clear  that  in  the  preceding  formula  /  may 
be  made  as  large  as  half  the  intersection  angle  of 
the  original  curve. 

If  it  be  desired  to  throw  the  middle  of  the  original 
simple  curve  out  along  a  radial  offset  for  a  distance 


THE  SIX-CHORD  SPIRAL  49 

hy  then,  72  being  half  the  intersection  angle  of  the 
original  curve, 

(25) 


Example  :  — 

Take  /  =  60°,  hence  /2  =  30° 

R0  =  955.4  (6°). 
p  =  4.4  ft.      h  =  0.5  ft. 

Then        RN  =  955.4  +  0.5  -  4'4^'5  -  919.3. 

Hence       RN  =  6°  14'  curve. 

Also,  vers^l--^-.  00479. 

7\  =  5°  36.6'. 

Hence  5°  36.6'  of  6°  14'  curve  are  to  be  replaced 
by  5°  36.6'  of  3°  07'  one-chord  approach. 

The  P.  C.  of  this  3°  07'  one-chord  will  be  back 
along  the  main  tangent  a  distance  = 

RN  sin  7\  -  (p  +  h)  cot  J  72  (26) 

from  the  original  P.  C., 

or  919.3  X  sin  5°  36.6'  -  (4.4  +  0.5)  cot  .15°  = 
71.60  ft. 

It  is  usually  best  to  run  such  curves  as  the  above 
from  both  ends,  making  the  junction  at  the  middle 
of  the  curve  or  on  the  radial  line  through  the 
vertex. 

If,  in  recentering  old  track,  the  best-fitting 
curve  should  merge  into  a  tangent  parallel  to,  and 
either  inside  (i)  or  outside  (o)  of  the  existing 


50  THE  SIX-CHORD  SPIRAL 

tangent  (which  is  to  be  maintained),  then  the 
amount  o  must  be  added  to,  and  the  amount  i 
subtracted  from,  p  in  formulas  (23)  to  (26). 

Example.  —  To  replace  part  of  a  4°  curve  that 
merges  into  a  parallel  tangent  2  ft.  outside  the 
existing  tangent,  by  enough  4°  30'  to  make  p, 
with  relation  to  the  existing  tangent,  =  6.62  ft. 

As  the  2-ft.  offset  o  is  outside,  equation  (23) 
becomes  : 

,       6.62  +  2       8.62       ._.. 
vers/  =  _  --  _  =  —  =  .0542 
KO  —  KN       159 

/  ==  18°  57' 

18°  57'  of  4°          =  473.75  ft.  to  be  replaced  by 
18°  57'  of  4°  30'  ==  421.11ft. 


Again,  vers  r,  =        -  jg  ~  .  0052 

Tl  =  5°  50.6'  of  4°  30'  =  129.87  ft. 

to  be  replaced  by  259.74  ft.  of  2°  15r  one-chord. 

The  P.  C.  of  this  one-chord  will  be  back  on  main 
tangent  from  the  original  P.  C.  4°  a  distance  of 

RN  sin  Tt  -  (p  +  o)  cot  i  /  = 
(1273.6  X  .1018)  -  5.992  (6.62  +  2)  =  78  ft. 

If  the  tangent  falls  inside  of  the  existing  tangent 
an  amount  i  (less  than  p)  of  2  ft.,  then 


Hence  /  =  13°  51',  and 

T,  =  5°  50.6'  as  before. 


THE  SIX-CHORD  SPIRAL  51 

The  distance  of  the  new  P.  C.  2°  15'  one-chord 
back  from  the  original  P.  C.  4°  will  be 

(1273.6  X  .1018) -8.233  (6.62  -  2)  =  91.61  ft. 

If  i  be  made  larger,  /  will  become  smaller  and 
the  4°  30'  curve  will  soon  be  too  short  to  serve  as 
the  base  for  a  2°  15'  one-chord  with  the  given 
value  of  p.  A  lighter  curve  must  then  be  taken, 
say  4°  20',  4°  10',  etc.,  until,  when  i  becomes  equal 
to  p,  the  4°  curve  is  connected  directly  with  the 
tangent  by  means  of  the  proper  length  of  2°  one- 
chord. 

When  i  exceeds  p,  a  curve  lighter  than  4°  must 
be  taken.  In  all  cases  the  total  angle  /  of  the 
terminal  branch  must  be  at  least  1J  times  7\  in 
order  to  make  room  for  the  six-chord,  and  at  least 
1.733  T1  for  the  track  parabola. 

In  addition  to  the  formulas  above  given,  the 
following  rule  for  shifting  the  P.  C.  C.  of  the  last 
arc  of  any  compound  curve  (without  changing  the 
degrees  of  curve)  in  order  to  offset  the  last  tangent 
parallel  to  itself,  is  of  constant  use. 

Rule  (Modified  from  Shunk's  "  Field  Book/'  page 
101) :  Divide  the  required  offset  by  the  difference 
of  the  radii,  and  call  the  quotient  Q;  call  the  nat. 
cosine  of  the  total  angle  of  the  located  last  arc  C. 
Then  either  Q  +  C  or  Q  —  C  will  be  the  nat.  cosine 
of  the  new  last  arc,  and  the  difference  between  the 
angle  whose  cos  =  C  and  the  angle  whose  cos  = 
Q  ±  C  gives  the  required  angular  shift  of  the  P.  C.  C. 


52  THE  SIX-CHORD  SPIRAL 

This  angular  shift  is  reduced  to  feet  according 
to  the  degree  of  curve  of  the  next-to-the-last  branch, 
and  on  which  it  must  be  used. 

It  is  evident  that 

1st.  To  offset  the  last  tangent  out  requires 
more  of  a  lighter  or  kss  of  a  sharper  final  arc. 

2d.  To  offset  the  last  tangent  in  requires  more 
of  a  sharper  or  less  of  a  lighter  final  arc. 

3d.  Less  final  arc  requires  more  cosine  ,  hence 
use  Q  +  C. 

4th.  Afore  final  arc  requires  less  cosine,  hence 
use  Q  —  C.  If  C  be  greater  than  Q,  use  C  -  Q. 

Example.  —  A30  compounds  into  a  5°,  which 
latter  has  a  total  angle  of  30°  22'.  It  is  desired  to 
throw  the  final  tangent  inward  34  ft. 

Here  R-r=  1,910  -  1,146  =  764,  and 


nat.  cos  30°  22'  =  .8628  =  C. 
nat,  cos  35°  05'  =  .8183  =  C  -  Q. 

In  this  case  the  tangent  is  to  be  thrown  in, 
hence  more  of  the  sharper  last  arc  (5°)  is  required. 
Therefore  use  C  -  Q  =  nat.  cos  35°  05'. 

35°  05'  -  30°  22'  =  4°  43' 

Since  more  sharper  last  arc  is  required,  the  P.  C.  C. 
must  be  moved  back  along  the  3°  curve  4°  43'  = 
157.22  ft. 

To  throw  the  tangent  out  34  ft.,  proceed  as 
follows  : 


THE  SIX-CHORD  SPIRAL  53 

.0445  +  .8628  =  .9073  =  nat.  cos  24°  52' . 
30°  22'  -  24°  52'  =  5°  30'. 

Here  the  P.  C.  C.  must  be  advanced  along  the 
3°  produced  5.5°  -  3°  =  183.33  ft. 

This  rule  may  be  used  to  shift  the  P.  C.  C.  of  a 
one-chord  spiral.  In  this  case  the  difference  of 
the  radii  =  2RM  —  RM  =  the  degree  of  the  main 
curve,  and  the  final  arc  of  one-chord  spiral  is 
always  the  lighter.  Hence,  to  offset  the  last 
tangent  out  requires  more  one-chord,  and  for  this 
use  Q  —  C.  To  offset  the  last  tangent  in  requires 
less  one-chord,  and  for  this  use  Q  +  C. 

Since,  in  this  case,  the  original  value  of  p  is 
always  known,  it  is  preferable  to  add  to  or  sub- 
tract from  p  (as  the  case  may  be)  the  required 
offset,  thus  forming  a  new  p  which  is  then  used  to 
determine  the  new  Tl  by  formula  (1). 

COMPOUND  CURVES. 

Space  may  be  made  at  the  P.  C.  C.  for  a  spiral 
between  the  two  members  of  a  compound  curve 
by  employing  one  of  the  following  methods: 

(1)  By  replacing  part  of  the  sharper  curve  with 
a  still  sharper  one. 

(2)  By  replacing  part  of  the  lighter  curve  with  a 
still  lighter  one. 

(3)  By  a  combination  of  (1)  and  (2),  preferably 
by  adding  as  many  minutes  to  the  degree  of  the 
sharper  curve  as  are  subtracted  from  the  degree  of 
the  lighter  one. 


54  THE  SIX-CHORD  SPIRAL 

By  the  third  method,  the  center  of  the  spiral 
practically  falls  on  the  original  P.  C.  C.,  and  the 
length  of  line  is  unchanged. 

First  Method.  —  When  sharpening  the  sharper 
curve  (of  degree  =  Ds)  to  DN  for  a  length  L^,  the 
original  lighter  curve  DL  must  be  produced  for  a 
distance  LL,  so  that  tangents  at  the  end  of  LN  of 
Ds  and  the  end  of  LL  of  DL  are  parallel  to  each 
other  and  p  feet  apart. 
(Inferiors:  S  —  Sharper;  L  =  Lighter;  N  =  New). 


Example.  —  A  2°  (DL)  and  8°  (Ds)  compound, 
and  it  is  required  to  insert  a  spiral,  p  being  taken 
at  3  ft.,  the  8°  to  be  changed  to  an  8°  30'  (DN)  ;  here 


LL    =  .2103  Stations  =  21.03  ft. 


LN    =  2.5237  Stations  =  252.37  ft. 

Hence,  the  beginning  of  the  new  8°  30'  (DN) 
curve  will  be  back  along  the  8°  curve  a  distance 
of  252.37  +  21.03  =  273.4  ft.  from  the  original 
P.  C.  C.,  and  the  resulting  condition  is  8°  30'  and 
2°  main  curves  parallel  to  each  other  at  a  point 


THE  SIX-CHORD  SPIRAL  55 

0$,)  21.03  ft.  from  the  original  P.  C.  C.  along  the 
original  2°  produced,  and  distant  apart  3  ft.;  re- 
quired to  connect  them  with  a 

oo  q/y  _i_  oo 

^-±       =  5°  15'  one-chord  spiral. 

This  5°  15'  curve  starts  from  the  2°  and  ends  on 
the  8°  30'  (or  vice  versa)  at  a  distance  from  S3 
(Fig.  3)  of 

dfa  (29) 

where  d  =  the  difference  between  the  degrees  of 
the  two  final  curves  (8°  30'  -  2°),  and  lt  =  length 
of  one-chord  in  stations  of  100  ft. 


Hence,  J  /1=  J-  -  =  .7284  Stations. 

'    .o7  X  o.o 

The  total  length  of  the  5°  15'  =  72.84  X  2  = 
145.7  ft.,  and  its  total  angle  =  7°  39'. 

(See  also  example  under  Fig.  3.) 

Second  Method.  —  When  lightening  the  lighter 
curve  DL  for  a  length  LN,  the  original  sharper 
curve  Ds  must  be  produced  a  distance  Ls  so  that 
tangents  at  the  end  of  LN  of  DL  and  the  end  of  Ls 
of  Ds  are  parallel  to  each  other  and  p  feet  apart. 


«» 

Example.—  A  2°   (Dz)   and   an   8°   (Ds)   com- 


56  THE  SIX-CHORD  SPIRAL 

pound,  and  it  is  required  to  insert  a  spiral,  p  being 
taken  at  3  ft.  and  the  2°  to  be  changed  to  a  1°  30' 
(DN). 
Here  V  =  L15  (8i2-  (^  315)=.  04423. 

L=21.03  ft. 


LN  =  252.37  ft. 

Hence  the  beginning  of  the  new  1°  30'  curve  will 
be  back  along  the  2°  curve  a  distance  of  252.37  + 
21.03  =  273.4  ft.  from  the  original  P.  C.  C.,  and 
the  resulting  condition  is  1°  30'  and  8°  main  curves 
parallel  to  each  other  at  a  point  (Sa)  21.03  ft.  from 
the  original  P.  C.  C.  along  the  original  8°  produced, 
and  distant  apart  3  ft. 

oo    I    -I  o  o/y 

Required   to  connect  them  with  a  - 

=  4°  45'  one-chord  spiral. 

This  4°  45'  curve  starts  from  the  8°  and  ends  on 
the  1°  30'  (or  vice  versa)  at  a  distance  from  S3 
(Fig.  3)  = 


where  d  is  the  difference  between  the  degrees  of 
the  two  final  curves  (8°  -  1°  30'),  and  Zt  =  length 
of  one-chord. 

Hence  }  I,  =  \  -  -  =  .7284  Stations,   or 

,o7  X  o.o 

72.84  ft.,  as  before. 


THE  SIX-CHORD  SPIRAL  57 

The  total  length  of  the  4°  45'  one-chord  =  72.84 
X  2  =  145.7,  and  its  total  angle  =  6°  55'. 

(See  also  Example  under  Fig.  3.) 

Third  Method.  —  When  both  the  sharper  curve 
is  sharpened  and  the  lighter  curve  lightened  by 
equal  amounts,  the  method  is  as  follows: 

Example.  —  A  2°  compounds  with  a  10°.  It  is 
desired  to  replace  150  ft.  of  the  2°  by  a  1°  30',  and 
150  ft.  of  the  10°  by  a  10°  30',  the  increase  and 
decrease  being  each  30'.  Here  the  line  will  be 
thrown  both  in  and  out  at  the  P.  C.  C.  for  a  dis- 
tance of 

J  p  =  .87  KLN\  (30) 

where  K  is  the  increase  or  decrease  expressed  in 
degrees  and  decimals,  and  LN  the  length  of  change 
of  each  curve  in  stations  of  100  feet.  In  this  case 

i  p  =  .87  X  .5  X  2.25  =  .98  ft.,  hence 
p  =  .98  x  2  =  1.96ft. 

The  resulting  condition  is  1°  30'  and  10°  30' 
curves  parallel  to  each  other  at  the  original  P.  C.  C. 
and  1.96  ft.  apart.  Required  to  connect  them 

.,,      10°  30' +  1°  30'  u    j      •    i 

with  a  -      — =  6°  one-chord  spiral. 

£t 

As  before,  this  6°  curve  starts  from  the  1°  30' 
and  ends  on  the  10°  30'  (or  vice  versa),  at  a  distance 
from  S3  (Fig.  3)  of 

P 


58  THE  SIX-CHORD  SPIRAL 

where  d  is  the  difference  between  the  degrees  of 
the  two  final  curves  and  /x  =  length  of  one-chord,  or 


v/'25  =  "5  stations' 


Hence  the  6°  will  have  a  total  length  of  .5  X  2  = 
100  ft.,  or  50  ft.  each  way  from  the  original  P.  C.  C. 

When  the  offset  p  is  given  and  also  the  increase 
and  decrease  of  the  degree  of  curve,  proceed  as 
follows  : 

Example.  —  A  2°  compounds  with  a  10°,  p  is  to 
be  taken  at  1.96  ft.,  and  1°  30'  and  10°  30'  curves 
used. 

Here 


where  LN  =  length  of  10°  30'  or  1°  30'  (to  be  used 
measured  from  original  P.  C.  C.),  p  =  principal  off- 
set, and  K  =  increase  or  decrease  of  degree  of  curve 
expressed  in  degrees  and  decimals. 

In  this  case  LN  =  .758  \  ~-  =  1.5  Stations  = 

.5 

150  ft.,  and  l±  is  found  as  above. 

These  methods  for  spiraling  compound  curves, 
though  approximate,  give  excellent  results  in 
practice. 

It  is  to  be  remembered  that  in  such  cases  the 
spiral  notes  are  not  used  to  replace  original  records; 
hence  there  is  no  real  need  of  absolute  accuracy. 


THE  SIX-CHORD  SPIRAL  59 

SPACE-SHIFTS  PRESERVING  THE  ORIGINAL 
LENGTH  OF  LINE. 

As  previously  indicated,  when  p  and  RM  are 
fixed,  the  one-chord,  six-chord  and  track  parabola 
all  give  the  same  length  of  line  between  common 
points  on  main  tangent  and  main  curve. 

Hence,  for  convenience  and  simplicity,  the  one- 
chord  will  be  considered  in  the  following  computa- 
tions : 

Given  two  tangents  intersecting  at  a  fixed  angle 
and  joined  by  simple  curves  of  various  radii. 

Call  the  distance  from  P.  C.  to  P.  T.  along  the 
tangents,  via  the  vertex,  the  tangent  route,  and  the 
distance  P.  C.  to  P.  T.,  via  the  curve,  the  curve 
route. 

Then  the  difference  between  the  tangent  and 
curve  routes  varies  in  direct  proportion  with  the 
radius  used. 

Further,  any  two  curve  routes  are  of  equal  length 
when  they  are  equally  less  than  the  tangent  route 
common  to  both. 

On  these  principles  ^the  following  solutions  are 
based : 

Example.  —  Given  a  4°  curve  for  70°  30',  to 
substitute  a  curve  with  spirals,  retaining  the  same 
length  of  line. 

Assume  a  terminal  angle  (7^)  of  3°  24'. 

First,  compute  the  elements  of  3°  24'  of  2°  one- 


60  THE  SIX-CHORD  SPIRAL 

chord  on  each  end  of  (70°  30'  -  6°  48'  =  )  63°  42' 

of  4°  main  curve. 

By  formula  (1)  p  =  vers  7\  X  RM  =  .00176  X 

1432.7  -  2.52  ft. 

By  formula  (10)  the  distance  from  the  apex  to 

P.  C.  of  one-chord  is  (RM  +  p)  tan  J  /  +  RM  sin  Tl 

(see  Fig.  1). 

RM  +  p=  1435.22  log  =  3.156918 

i  /  =35°  15'       log  tan  =  9.849254 

GA          =  1014.31  log  =  3.006172 

RM          =  1432.7  log  =  3.156151 

Tt  =  3°  24'          log  sin  =  8.773101 

GH          =  84.97  log  =  1.929252 

AH  =  GA  +  GH  =    1099.28 
Hence  tangent  route  = 

2  (1014.31  +  84.97)  =  2198.56 
By  the  curve  route  there  is 
6°  48'  of  2°  =  340  ft.,  and 
63°  42'  of  4°  =  1592.5  ft. :    total      =  1932.50 
Tangent  route  less  curve  route         =     266.06 
Now,  to  preserve  the  original  length  of  line,  this 
difference  must  be  reduced  by  shortening  the  radii 
until  it  equals  the  original  difference  between  the 
tangent  and  curve  routes.     The  latter  is  calculated 
thus: 

RM  =  1432.7  log  =  3.156151 

i  7  =  35°  15'   log  tan  =  9.849254 
EA  =  1012.52         log  =  3.005405 


THE  SIX-CHORD  SPIRAL  61 

1012.52  X  2  =  2025.04 

Length  4°  for  70°  30'    =  1762.50 
Tangent  route  less  orig- 
inal curve  route        =      262.54 


Then  262.54  : 266.06  :  :4°:  4.054°  (=  4°  03.2'). 
Hence  each  terminal  curve  will  consist  of  3°  24' 
of  2°  01.6'  curve. 
The  new  apex  distance  A  H  will  be 

266.06  :  262.54  ::  1099.28  : 1084.74. 

Similarly,  the  new  p  =  2.487. 

In  working  out  these  proportions  use  logarithms. 

For  running  in  such  a  curve  as  a  4.054°,  the 
decimal  vernier  (formerly  supplied  on  transits  by 
Young  &  Sons,  of  Philadelphia)  is  a  great  con- 
venience. These  instruments  had  one  decimal 
vernier,  the  opposite  one  being  of  the  usual  sex- 
agesimal form. 

If,  instead  of  the  terminal  angle  3°  24',  the  final 
value  of  p  =  2.487  be  given,  proceed  as  follows : 

1st.  Calculate  the  difference  between  the  tan- 
gent and  original  curve  routes;  call  this  A. 

2d.  Multiply  twice  p  by  the  tangent  of  half  the 
whole  intersection  angle;  call  this  product  B. 
Then 

A  :A  +  B  ::D0  :DN 

where          D0  =  degree  of  original  main  curve, 

DN  =  degree  of  new  main  curve, 
\  DN  =  degree  of  new  one-chord. 


62  THE  SIX-CHORD  SPIRAL 

Example.  —  Given  a  4°  curve  for  70°  30',  to 
substitute  a  curve  with  spirals,  retaining  the  same 
length  of  line  and  assuming  p  =  2.487. 

From  the  preceding  example,  tangent  route  less 
the  original  curve  route  =  262.54  =  A. 

Also,  2p       =  4.974  log  =  0.696706 

i  /        -  35°  15'    log  tan  =  9.849254 

B         =  3.52  log  =  0.545960 

A  +  B  =  266.06 

Then  262.54  :  266.06  ::  4°  :  4.054°  (=  4°  03.2'). 

Other  elements  of  the  curves  are  found  from 
formulas  (1)  and  (10),  as  before. 

Similarly,  spirals  may  be  inserted  at  the  ends 
and  between  the  members  of  a  compound  curve, 
while  preserving  the  original  length  of  line.  This 
is  shown  by  the  following  example,  which  also 
serves  as  a  general  review. 

Given  a  compound  curve,  as  follows  (see  Fig.  4) : 

Sta.  10  P.  C.         4°  R  for  32°  of  angle  =  b. 

Sta.  18  P.  C.  C.  10°  R  for  50°  of  angle  =  c. 

Sta.  23  P.  T. 

Required  to  insert  spirals  at  P.  C.,  P.  C.  C.,  and 
P.  T.  without  changing  length  of  line. 

Maximum  speed  on  10°  curve  =  31.6  miles  per 
hour,  which  will  also  be  taken  on  the  4°. 

By  Rule  2,  length  of  one-chord  spiral  equals 
elevation  in  tenths  of  feet  multiplied  by  speed. 
Hence,  from  Table  III: 


THE  SIX-CHORD  SPIRAL  63 

Length  of  one-chord  for  10°  curve  =  31.6  X 
5  =  158.0  ft. 

Length  of  one-chord  for  4°  curve  =  31.6  X 
2  =  63.2  ft. 

Length  of  one-chord  between  4°  and  10°  =  that 
for  10  -  4  =  6°. 

Length  of  one-chord  for  6°  curve  =  31.6  X  3 
=  94.8  ft. 

In  this  example  radius  =  5730  -r-  degree  of 
curve, 

The  values  of  p  are  as  follows:  Since  p  =  RM  X 
vers  Tlf 

Terminal  ang.  7\for  10°  =  7°  54',  p  =  5.44  =  P. 
Terminal  ang.  7\  for  4°  =  1°  16',  p  =  .34  =  p. 
Terminal  ang.  7\for  6°  =  2°  51',  p  =  1.18  =  p4 

By  formula  (13): 

/   £  44  \ 

-  -1.18    cos  32°  -0.34 

=  5.893. 


sm  82° 
By  formula  (14)  : 

FW  =  (5.44  X  tan  50°)  -  5.893  -  .590. 
TL   =  ML  X  sin  b  =  EH  X  sin  b. 

EH  =-^--Pl  =  7.283. 
cos  C 

Hence  7.283  X  sin  32°  =  3.860  =  TL. 
Next  calculate  the  effect  of  the  shift  GV  meas- 
ured along  LT  produced. 


64  THE  SIX-CHORD  SPIRAL 

This  will  equal  WN  X  cos  (b  +  c) 
=  5.893  X  cos  82°  =  0.820  =  TZ. 
To  this  add  TL        =  3.860. 
Hence  shift  LZ        =  4.680. 


The  notes  of  a  new  curve  with  one-chord  spirals 
inserted,    but   retaining   the   original   degrees   of 
curve,  would  be  as  follows : 
P.  C.  4°  =  point  L  (Fig.  4)  =  Station      10  +00. 
Less  LZ  04.68 

Point  Z  (Fig.  4)  =  9  +  95.32 

Deduct  GH  (Fig.  1)  =  1432.5  X  sin 

1°  16',  [from  (8)]  31.60 

9  +  63.72 
Then 

9  +  63.72     P.  C.    2°  one-chord  for         1°  16' 
+  63.20 


10  +  26.92  P.  C.  C.    4°  main  curve  for    28°  50' 
7  +  20.83 

17  +  47.75  P.  C.  C.    7°  one-chord  for        6°  38' 

94.80 

18  +  42.55  P.  C.  C.  10°  main  curve  for    37°  22' 
3  +  73.67 

22  +  16.22  P.  C.  C.    5°  one-chord  for  _     7°  54' 
.1  +  58 

23  +  74.22  P.  T.  82°  00' 

Thus  far  the  procedure  has  been  the  same  as 
though  the  change  was  to  be  made  in  the  first  line 


THE  SIX-CHORD  SPIRAL  65 

prior  to  construction.  It  remains  to  reduce  the 
above  to  a  similar  figure,  having  the  same  length 
between  common  points  as  the  original  line.  For 
this  purpose  first  calculate  the  original  apex  dis- 
tances, A  to  L  and  F  (Fig.  4),  from  the  following 
formulas,  A  being  the  intersection  of  the  main  tan- 
gents through  L  and  F  produced  (not  shown  in 
figure)  : 


AF  =  fl.  tan  \I  + 


, 

sin/ 

~  fi')fvers 
sin/ 


where  R2  is  the  larger  radius  =  1432.5 

Rl  is  the  smaller  radius  =    573.0 

/  =  the  grand  total  angle     =        82° 
/t  =  the  Rt  total  angle  c  50° 

/2  =  the  R2  total  angle  b  32° 

AL  is  on  the  side  of  the  lighter  curve  and  A  F 
on  that  of  the  sharper. 
Hence  AL  =    935.21 

AF  =    629.96 

Original  tangent  route  =  1565.17 

Original  curve  route  (2300  -  1000)  =  1300.00 
Difference  =    265.17 

The  new  curve  route  =  23  +  74.22 
less         9  +  63.72 
or       14  +  10.50 
The  new  tangent  route  is  obtained  thus: 


66  THE  SIX-CHORD  SPIRAL 

1st.   Distance  from  apex  to  P.  C.  2°  one-chord. 
Original  tangent   =  AL  =  935.21 
LZ  =      4.68 

R  4°  X  sin  1°  16'  31.60    971.49 

2d.   Distance  from  apex  to  P.  T.  5°. 
Original  tangent  AF   =  629.96 
FW   •          .59 

R  10°  X  sin  7°  54'  78.75      709.30 

Total  via  tangent  route  1680.79 

Total  via  new  curve  route  1410.50 

Difference  270.29 

Hence  required  ratio  = 

270.29 

=  1.0193,  or,  inversely, 


265.17 
265.17 
270.29 


=    .9810. 


Hence    4°  becomes  4°  X  1.0193  =     4°  04.6', 
and  10°  becomes         10.1930  =  10°  11.6'. 
The  new  tangents  will  be 

971.49  X  .981  =     953.03  for  AL. 
709.30  X  .981  =      695.82  for  AF. 
Final  tangent  route     =  1648.85 
971.49  -  953.03  =       18.46 
And  the  final  alinement  notes  will  read : 
9  +  63.72  trial  P.  C.  2°. 

18.46 

9  +  82.18  P.  C.        2°  02.3'  for    1°  16' 
62. 


THE  SIX-CHORD  SPIRAL  67 

10  +  44.18  P.  C.  C.    4°  04.6'  for  28°  50' 
7  +  07.13. 

17  +  51.31  P.  C.  C.    7°  08.1'  for    6°  38' 

93. 

18  +  44.31  P.  C.  C.  10°  11.6'  for  37°  22' 
3  +  66.57. 

22  +  10.88  P.  C.  C.    5°  05.8'  for  7°  54' 
1  +  55.  82°  00' 


23  +  65.88  P.  T. 
-  9  +  82.18 P.O. 
13  +  83.70  =  final  curve  route. 
16  +  48.85  =  final  tangent  route. 

265.15  =  final  difference. 

265.17  =  original  difference. 

The  quantities  added  in  the  above  tabulation 
are  those  in  the  preceding  alinement  table  X  .981. 

Having  thus  computed  the  required  one-chords, 
the  corresponding  six-chord  spirals  or  track  para- 
bolas may  be  traced,  as  previously  shown. 

If  no  spiral  be  required  between  the  two  mem- 
bers of  the  compound  curve,  make  p  =  zero  in 
formulas  (11)  to  (19). 

The  spiral  at  the  P.  C.  C.  may  be  subsequently 
run  in  by  the  methods  of  formulas  (30)  and  (31). 

For  the  case  of  a  simple  curve  terminating  in 
unequal  spirals  use  formulas  (11)  to  (19),  making 


68  THE  SIX-CHORD  SPIRAL 


b  +  c      = 

p  t  =  zero. 

P  and  p  =  their  assumed  values. 

Calculations  such  as  the  preceding  should  be 
made  in  the  office  after  a  careful  survey  of  the 
existing  track  has  been  made. 

On  the  plat  of  this  survey  the  most  suitable 
points  for  widening  cuts  and  fills  to  make  room 
for  spirals,  must  be  noted. 

Each  division,  at  least,  of  the  road  should  be 
treated  by  one  experienced  man.  This  will  insure 
uniform  and  consistent  spiraling. 

The  whole  should  be  formally  approved  by  the 
highest  available  operating  officer  before  being 
traced  on  the  ground. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


